:: JORDAN1H semantic presentation

begin

registration
let D be ( ( non empty with_non-empty_element ) ( non empty with_non-empty_element ) set ) ;
cluster non empty Relation-like non-empty NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined D : ( ( non empty with_non-empty_element ) ( non empty with_non-empty_element ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of D : ( ( non empty with_non-empty_element ) ( non empty with_non-empty_element ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() FinSequence-yielding finite-support for ( ( ) ( ) FinSequence of D : ( ( non empty with_non-empty_element ) ( non empty with_non-empty_element ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of D : ( ( non empty with_non-empty_element ) ( non empty with_non-empty_element ) set ) ) ) ;
end;

registration
let F be ( ( Relation-like non-empty Function-like Function-yielding ) ( Relation-like non-empty Function-like Function-yielding V81() ) Function) ;
cluster rngs F : ( ( Relation-like non-empty Function-like Function-yielding ) ( Relation-like non-empty Function-like Function-yielding V81() ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) -> Relation-like non-empty Function-like ;
end;

theorem :: JORDAN1H:1
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) <> q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) holds
p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) in Cl ((LSeg (p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) )) : ( ( ) ( non empty functional closed boundary connected bounded compact convex ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) \ {p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) } : ( ( ) ( functional finite V32() ) set ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1H:2
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) <> q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) holds
Cl ((LSeg (p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) )) : ( ( ) ( non empty functional closed boundary connected bounded compact convex ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) \ {p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) } : ( ( ) ( functional finite V32() ) set ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) = LSeg (p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( non empty functional closed boundary connected bounded compact convex ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1H:3
for S being ( ( ) ( functional ) Subset of )
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) st p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) <> q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) & (LSeg (p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) )) : ( ( ) ( non empty functional closed boundary connected bounded compact convex ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) \ {p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) } : ( ( ) ( functional finite V32() ) set ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) c= S : ( ( ) ( functional ) Subset of ) holds
LSeg (p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( non empty functional closed boundary connected bounded compact convex ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) c= Cl S : ( ( ) ( functional ) Subset of ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ;

begin

definition
func RealOrd -> ( ( ) ( Relation-like REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued V59() V60() V61() ) Relation of ) equals :: JORDAN1H:def 1
 { [r : ( ( ) ( V11() real ext-real ) Real) ,s : ( ( ) ( V11() real ext-real ) Real) ] : ( ( ) ( ) set ) where r, s is ( ( ) ( V11() real ext-real ) Real) : r : ( ( ) ( V11() real ext-real ) Real) <= s : ( ( ) ( V11() real ext-real ) Real) } ;
end;

theorem :: JORDAN1H:4
for r, s being ( ( ) ( V11() real ext-real ) Real) st [r : ( ( ) ( V11() real ext-real ) Real) ,s : ( ( ) ( V11() real ext-real ) Real) ] : ( ( ) ( ) set ) in RealOrd : ( ( ) ( Relation-like REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued V59() V60() V61() ) Relation of ) holds
r : ( ( ) ( V11() real ext-real ) Real) <= s : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: JORDAN1H:5
field RealOrd : ( ( ) ( Relation-like REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued V59() V60() V61() ) Relation of ) : ( ( ) ( ) set ) = REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ;

registration
cluster RealOrd : ( ( ) ( Relation-like REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued V59() V60() V61() ) Relation of ) -> total reflexive antisymmetric transitive being_linear-order ;
end;

theorem :: JORDAN1H:6
RealOrd : ( ( ) ( Relation-like REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued total reflexive antisymmetric transitive V59() V60() V61() being_linear-order ) Relation of ) linearly_orders REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ;

theorem :: JORDAN1H:7
for A being ( ( finite ) ( finite V69() V70() V71() ) Subset of ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) holds SgmX (RealOrd : ( ( ) ( Relation-like REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued total reflexive antisymmetric transitive V59() V60() V61() being_linear-order ) Relation of ) ,A : ( ( finite ) ( finite V69() V70() V71() ) Subset of ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) is increasing ;

theorem :: JORDAN1H:8
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) )
for A being ( ( finite ) ( finite V69() V70() V71() ) Subset of ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) st A : ( ( finite ) ( finite V69() V70() V71() ) Subset of ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) = rng f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) : ( ( ) ( finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) holds
SgmX (RealOrd : ( ( ) ( Relation-like REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued total reflexive antisymmetric transitive V59() V60() V61() being_linear-order ) Relation of ) ,A : ( ( finite ) ( finite V69() V70() V71() ) Subset of ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) = Incr f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) : ( ( increasing ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like V59() V60() V61() increasing V65() finite-support ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ;

registration
let A be ( ( finite ) ( finite V69() V70() V71() ) Subset of ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ;
cluster SgmX (RealOrd : ( ( ) ( Relation-like REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued total reflexive antisymmetric transitive V59() V60() V61() being_linear-order ) Relation of ) ,A : ( ( finite ) ( finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) -> increasing ;
end;

theorem :: JORDAN1H:9
for X being ( ( non empty ) ( non empty ) set )
for A being ( ( finite ) ( finite ) Subset of ( ( ) ( ) set ) )
for R being ( ( total reflexive antisymmetric transitive being_linear-order ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued total reflexive antisymmetric transitive being_linear-order ) Order of ) holds len (SgmX (R : ( ( total reflexive antisymmetric transitive being_linear-order ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued total reflexive antisymmetric transitive being_linear-order ) Order of ) ,A : ( ( finite ) ( finite ) Subset of ( ( ) ( ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) = card A : ( ( finite ) ( finite ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of omega : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) set ) ) ;

begin

theorem :: JORDAN1H:10
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) holds X_axis f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) = proj1 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) * f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like finite V59() V60() V61() finite-support ) Element of bool [:NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ;

theorem :: JORDAN1H:11
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) holds Y_axis f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) = proj2 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) * f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like finite V59() V60() V61() finite-support ) Element of bool [:NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ;

definition
let D be ( ( non empty ) ( non empty ) set ) ;
let M be ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined D : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of D : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() FinSequence-yielding finite-support ) FinSequence of D : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of D : ( ( non empty ) ( non empty ) set ) ) ) ;
:: original: Values
redefine func Values M -> ( ( ) ( ) Subset of ( ( ) ( ) set ) ) ;
end;

registration
let D be ( ( non empty with_non-empty_elements ) ( non empty with_non-empty_elements non empty-membered ) set ) ;
let M be ( ( non empty non-empty ) ( non empty Relation-like non-empty NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined D : ( ( non empty with_non-empty_elements ) ( non empty with_non-empty_elements non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of D : ( ( non empty with_non-empty_elements ) ( non empty with_non-empty_elements non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() FinSequence-yielding finite-support ) FinSequence of D : ( ( non empty with_non-empty_elements ) ( non empty with_non-empty_elements non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of D : ( ( non empty with_non-empty_elements ) ( non empty with_non-empty_elements non empty-membered ) set ) ) ) ;
cluster Values M : ( ( non empty non-empty ) ( non empty Relation-like non-empty NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined D : ( ( non empty with_non-empty_elements ) ( non empty with_non-empty_elements non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of D : ( ( non empty with_non-empty_elements ) ( non empty with_non-empty_elements non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() FinSequence-yielding finite-support ) FinSequence of D : ( ( non empty with_non-empty_elements ) ( non empty with_non-empty_elements non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of D : ( ( non empty with_non-empty_elements ) ( non empty with_non-empty_elements non empty-membered ) set ) ) ) : ( ( ) ( finite ) set ) -> non empty ;
end;

theorem :: JORDAN1H:12
for D being ( ( non empty ) ( non empty ) set )
for M being ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular FinSequence-yielding finite-support ) Matrix of ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of D : ( ( non empty ) ( non empty ) set ) ) )
for i being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) in Seg (width M : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular FinSequence-yielding finite-support ) Matrix of ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( finite width b2 : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular FinSequence-yielding finite-support ) Matrix of ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element V69() V70() V71() V72() V73() V74() ) Element of bool NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) holds
rng (Col (M : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular FinSequence-yielding finite-support ) Matrix of ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support ) Element of (len b2 : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular FinSequence-yielding finite-support ) Matrix of ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -tuples_on b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( finite ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) c= Values M : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular FinSequence-yielding finite-support ) Matrix of ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( finite ) Subset of ( ( ) ( ) set ) ) ;

theorem :: JORDAN1H:13
for D being ( ( non empty ) ( non empty ) set )
for M being ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular FinSequence-yielding finite-support ) Matrix of ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of D : ( ( non empty ) ( non empty ) set ) ) )
for i being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) in dom M : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular FinSequence-yielding finite-support ) Matrix of ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( finite V69() V70() V71() V72() V73() V74() ) Element of bool NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) holds
rng (Line (M : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular FinSequence-yielding finite-support ) Matrix of ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support ) Element of (width b2 : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular FinSequence-yielding finite-support ) Matrix of ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -tuples_on b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( finite ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) c= Values M : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular FinSequence-yielding finite-support ) Matrix of ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( finite ) Subset of ( ( ) ( ) set ) ) ;

theorem :: JORDAN1H:14
for G being ( ( V15() tabular X_increasing-in-column ) ( Relation-like V15() NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_increasing-in-column FinSequence-yielding finite-support ) Matrix of ) holds len G : ( ( V15() tabular X_increasing-in-column ) ( Relation-like V15() NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_increasing-in-column FinSequence-yielding finite-support ) Matrix of ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= card (proj1 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (Values G : ( ( V15() tabular X_increasing-in-column ) ( Relation-like V15() NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_increasing-in-column FinSequence-yielding finite-support ) Matrix of ) ) : ( ( ) ( functional finite closed bounded compact ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of omega : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) set ) ) ;

theorem :: JORDAN1H:15
for G being ( ( tabular X_equal-in-line ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line FinSequence-yielding finite-support ) Matrix of ) holds card (proj1 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (Values G : ( ( tabular X_equal-in-line ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line FinSequence-yielding finite-support ) Matrix of ) ) : ( ( ) ( functional finite closed bounded compact ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of omega : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) set ) ) <= len G : ( ( tabular X_equal-in-line ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line FinSequence-yielding finite-support ) Matrix of ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ;

theorem :: JORDAN1H:16
for G being ( ( V15() tabular X_equal-in-line X_increasing-in-column ) ( Relation-like V15() NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line X_increasing-in-column FinSequence-yielding finite-support ) Matrix of ) holds len G : ( ( V15() tabular X_equal-in-line X_increasing-in-column ) ( Relation-like V15() NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line X_increasing-in-column FinSequence-yielding finite-support ) Matrix of ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) = card (proj1 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (Values G : ( ( V15() tabular X_equal-in-line X_increasing-in-column ) ( Relation-like V15() NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line X_increasing-in-column FinSequence-yielding finite-support ) Matrix of ) ) : ( ( ) ( functional finite closed bounded compact ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of omega : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) set ) ) ;

theorem :: JORDAN1H:17
for G being ( ( V15() tabular Y_increasing-in-line ) ( Relation-like V15() NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular Y_increasing-in-line FinSequence-yielding finite-support ) Matrix of ) holds width G : ( ( V15() tabular Y_increasing-in-line ) ( Relation-like V15() NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular Y_increasing-in-line FinSequence-yielding finite-support ) Matrix of ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= card (proj2 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (Values G : ( ( V15() tabular Y_increasing-in-line ) ( Relation-like V15() NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular Y_increasing-in-line FinSequence-yielding finite-support ) Matrix of ) ) : ( ( ) ( functional finite closed bounded compact ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of omega : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) set ) ) ;

theorem :: JORDAN1H:18
for G being ( ( V15() tabular Y_equal-in-column ) ( Relation-like V15() NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular Y_equal-in-column FinSequence-yielding finite-support ) Matrix of ) holds card (proj2 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (Values G : ( ( V15() tabular Y_equal-in-column ) ( Relation-like V15() NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular Y_equal-in-column FinSequence-yielding finite-support ) Matrix of ) ) : ( ( ) ( functional finite closed bounded compact ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of omega : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) set ) ) <= width G : ( ( V15() tabular Y_equal-in-column ) ( Relation-like V15() NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular Y_equal-in-column FinSequence-yielding finite-support ) Matrix of ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ;

theorem :: JORDAN1H:19
for G being ( ( V15() tabular Y_equal-in-column Y_increasing-in-line ) ( Relation-like V15() NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular Y_equal-in-column Y_increasing-in-line FinSequence-yielding finite-support ) Matrix of ) holds width G : ( ( V15() tabular Y_equal-in-column Y_increasing-in-line ) ( Relation-like V15() NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular Y_equal-in-column Y_increasing-in-line FinSequence-yielding finite-support ) Matrix of ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) = card (proj2 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (Values G : ( ( V15() tabular Y_equal-in-column Y_increasing-in-line ) ( Relation-like V15() NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular Y_equal-in-column Y_increasing-in-line FinSequence-yielding finite-support ) Matrix of ) ) : ( ( ) ( functional finite closed bounded compact ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of omega : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) set ) ) ;

begin

theorem :: JORDAN1H:20
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board)
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) holds
for k being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) c= left_cell (f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1H:21
for k being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for f being ( ( non empty circular special unfolded s.c.c. standard ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) st 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= len f : ( ( non empty circular special unfolded s.c.c. standard ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
left_cell (f : ( ( non empty circular special unfolded s.c.c. standard ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) ,k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(GoB f : ( ( non empty circular special unfolded s.c.c. standard ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) ) : ( ( tabular ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) = left_cell (f : ( ( non empty circular special unfolded s.c.c. standard ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) ,k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1H:22
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board)
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) holds
for k being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
LSeg (f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) c= right_cell (f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1H:23
for k being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for f being ( ( non empty circular special unfolded s.c.c. standard ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) st 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= len f : ( ( non empty circular special unfolded s.c.c. standard ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
right_cell (f : ( ( non empty circular special unfolded s.c.c. standard ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) ,k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(GoB f : ( ( non empty circular special unfolded s.c.c. standard ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) ) : ( ( tabular ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) = right_cell (f : ( ( non empty circular special unfolded s.c.c. standard ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) ,k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1H:24
for P being ( ( ) ( functional ) Subset of )
for f being ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) holds
( not P : ( ( ) ( functional ) Subset of ) is_a_component_of (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) ) : ( ( ) ( non empty functional closed boundary bounded compact non horizontal non vertical ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ` : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) or P : ( ( ) ( functional ) Subset of ) = RightComp f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) : ( ( ) ( non empty functional open connected V199( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) or P : ( ( ) ( functional ) Subset of ) = LeftComp f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) : ( ( ) ( non empty functional open connected V199( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1H:25
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board)
for f being ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) st f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) 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 ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) holds
for k being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= len f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
( Int (right_cell (f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) ,k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) )) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) c= RightComp f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) : ( ( ) ( non empty functional open connected V199( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) & Int (left_cell (f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) ,k : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) )) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) c= LeftComp f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) : ( ( ) ( non empty functional open connected V199( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1H:26
for i1, j1, i2, j2 being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) st [i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) : ( ( ) ( ) set ) & [i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) : ( ( ) ( ) set ) & G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) * (i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) * (i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) holds
( i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) = i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) = j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) ;

theorem :: JORDAN1H:27
for i1, i2 being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) )
for M being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) st f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on M : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) holds
mid (f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on M : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( Relation-like non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ;

registration
cluster non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column -> non empty non-empty non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column for ( ( ) ( ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ;
end;

theorem :: JORDAN1H:28
for i being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) st 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
(SgmX (RealOrd : ( ( ) ( Relation-like REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued total reflexive antisymmetric transitive V59() V60() V61() being_linear-order ) Relation of ) ,(proj1 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (Values G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( functional finite closed bounded compact ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like V59() V60() V61() increasing V65() finite-support ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) . i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) = (G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) * (i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) `1 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ;

theorem :: JORDAN1H:29
for j being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) st 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
(SgmX (RealOrd : ( ( ) ( Relation-like REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued total reflexive antisymmetric transitive V59() V60() V61() being_linear-order ) Relation of ) ,(proj2 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (Values G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( functional finite closed bounded compact ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like V59() V60() V61() increasing V65() finite-support ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) . j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) = (G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) * (1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ;

theorem :: JORDAN1H:30
for f being ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) st f : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) & ex i being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st
( [1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) : ( ( ) ( ) set ) & G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) * (1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) in rng f : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( non empty functional finite closed bounded compact ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ) & ex i being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st
( [(len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) : ( ( ) ( ) set ) & G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) * ((len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) in rng f : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( non empty functional finite closed bounded compact ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ) holds
proj1 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (rng f : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( non empty functional finite closed bounded compact ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) = proj1 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (Values G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( functional finite closed bounded compact ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ;

theorem :: JORDAN1H:31
for f being ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) st f : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) & ex i being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st
( [i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) : ( ( ) ( ) set ) & G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) * (i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) in rng f : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( non empty functional finite closed bounded compact ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ) & ex i being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st
( [i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) : ( ( ) ( ) set ) & G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) * (i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) in rng f : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( non empty functional finite closed bounded compact ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ) holds
proj2 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (rng f : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( non empty functional finite closed bounded compact ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) = proj2 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (Values G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( functional finite closed bounded compact ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ;

registration
let G be ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ;
cluster Values G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( finite ) set ) -> non empty ;
end;

theorem :: JORDAN1H:32
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) holds G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) = GoB ((SgmX (RealOrd : ( ( ) ( Relation-like REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued total reflexive antisymmetric transitive V59() V60() V61() being_linear-order ) Relation of ) ,(proj1 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (Values G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( non empty functional finite closed bounded compact ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( non empty finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )) : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like V59() V60() V61() increasing V65() finite-support ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ,(SgmX (RealOrd : ( ( ) ( Relation-like REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued total reflexive antisymmetric transitive V59() V60() V61() being_linear-order ) Relation of ) ,(proj2 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (Values G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( non empty functional finite closed bounded compact ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( non empty finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )) : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like V59() V60() V61() increasing V65() finite-support ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ;

theorem :: JORDAN1H:33
for f being ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) st proj1 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (rng f : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( non empty functional finite closed bounded compact ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) = proj1 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (Values G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( non empty functional finite closed bounded compact ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) & proj2 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (rng f : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( non empty functional finite closed bounded compact ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) = proj2 : ( ( Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) ) ( Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) -valued Function-like V43( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) , REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) V59() V60() V61() ) Element of bool [: the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ,REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) :] : ( ( ) ( non empty non trivial Relation-like non finite non empty-membered V59() V60() V61() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) .: (Values G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( non empty functional finite closed bounded compact ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty finite V69() V70() V71() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) holds
G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) = GoB f : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ;

theorem :: JORDAN1H:34
for f being ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) )
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) st f : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) & ex i being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st
( [1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) : ( ( ) ( ) set ) & G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) * (1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) in rng f : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( non empty functional finite closed bounded compact ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ) & ex i being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st
( [i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) : ( ( ) ( ) set ) & G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) * (i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) in rng f : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( non empty functional finite closed bounded compact ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ) & ex i being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st
( [(len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) : ( ( ) ( ) set ) & G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) * ((len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) in rng f : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( non empty functional finite closed bounded compact ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ) & ex i being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st
( [i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) : ( ( ) ( ) set ) & G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) * (i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) in rng f : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( non empty functional finite closed bounded compact ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ) holds
G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) Go-board) = GoB f : ( ( non empty ) ( non empty Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ;

begin

theorem :: JORDAN1H:35
for i, n, m being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
[\(((i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) - 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( V11() real integer ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) |^ (n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' m : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) + 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) /] : ( ( integer ) ( V11() real integer ext-real ) set ) is ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ;

theorem :: JORDAN1H:36
for m, n, i being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for C being ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) st m : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= len (Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
( 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= [\(((i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) - 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( V11() real integer ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) |^ (n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' m : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) + 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) /] : ( ( integer ) ( V11() real integer ext-real ) set ) & [\(((i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) - 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( V11() real integer ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) |^ (n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' m : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) + 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) /] : ( ( integer ) ( V11() real integer ext-real ) set ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( V11() real integer ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) <= len (Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) ;

theorem :: JORDAN1H:37
for m, n, i, j being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for C being ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) st m : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= len (Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= width (Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
ex i1, j1 being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st
( i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) = [\(((i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) - 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( V11() real integer ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) |^ (n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' m : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) + 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) /] : ( ( integer ) ( V11() real integer ext-real ) set ) & j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) = [\(((j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) - 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( V11() real integer ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) / (2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) |^ (n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' m : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) + 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) ) /] : ( ( integer ) ( V11() real integer ext-real ) set ) & cell ((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) c= cell ((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ,i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1H:38
for m, n, i, j being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for C being ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) st m : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= len (Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= width (Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
ex i1, j1 being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= len (Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= width (Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & cell ((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) c= cell ((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ,i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1H:39
for P being ( ( ) ( functional ) Subset of ) st P : ( ( ) ( functional ) Subset of ) is bounded holds
not UBD P : ( ( ) ( functional ) Subset of ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) is bounded ;

theorem :: JORDAN1H:40
for p being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) )
for f being ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) st Rotate (f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) ,p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Point of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is clockwise_oriented holds
f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) is clockwise_oriented ;

theorem :: JORDAN1H:41
for f being ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) st LeftComp f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) : ( ( ) ( non empty functional open connected V199( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) = UBD (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) ) : ( ( ) ( non empty functional closed boundary bounded compact non horizontal non vertical ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional open connected V199( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) holds
f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) is clockwise_oriented ;

begin

theorem :: JORDAN1H:42
for f being ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) holds (Cl (LeftComp f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) ) : ( ( ) ( non empty functional open connected V199( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ` : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) = RightComp f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) : ( ( ) ( non empty functional open connected V199( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1H:43
for f being ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) holds (Cl (RightComp f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) ) : ( ( ) ( non empty functional open connected V199( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ` : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) = LeftComp f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard finite-support ) special_circular_sequence) : ( ( ) ( non empty functional open connected V199( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1H:44
for n being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for C being ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) st C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) is connected holds
GoB (Cage (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard clockwise_oriented finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) = Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ;

theorem :: JORDAN1H:45
for n being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for C being ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) st C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) is connected holds
N-min C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) in right_cell ((Cage (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard clockwise_oriented finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1H:46
for i, j being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for C being ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) st C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) is connected & i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
L~ (Cage (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard clockwise_oriented finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( non empty functional closed boundary bounded compact non horizontal non vertical ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) c= Cl (RightComp (Cage (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard clockwise_oriented finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( non empty functional open connected V199( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1H:47
for i, j being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for C being ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) st C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) is connected & i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
LeftComp (Cage (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard clockwise_oriented finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( non empty functional open connected V199( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) c= LeftComp (Cage (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard clockwise_oriented finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( non empty functional open connected V199( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1H:48
for i, j being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for C being ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) st C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) is connected & i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
RightComp (Cage (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard clockwise_oriented finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( non empty functional open connected V199( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) c= RightComp (Cage (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty non trivial Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like non constant finite FinSequence-like FinSubsequence-like non empty-membered Function-yielding V81() circular special unfolded s.c.c. standard clockwise_oriented finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( non empty functional open connected V199( TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) ;

begin

definition
let C be ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ;
let n be ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ;
func X-SpanStart (C,n) -> ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) equals :: JORDAN1H:def 2
(2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) |^ (n : ( ( non empty-membered ) ( non empty non empty-membered ) Element of bool K563(C : ( ( non empty with_non-empty_element ) ( non empty with_non-empty_element ) set ) ) : ( ( V218() ) ( V218() ) set ) : ( ( ) ( ) set ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ;
end;

theorem :: JORDAN1H:49
for n being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for C being ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) holds
( 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) < X-SpanStart (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & X-SpanStart (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) < len (Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) ;

theorem :: JORDAN1H:50
for n being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for C being ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) holds
( 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) <= (X-SpanStart (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & (X-SpanStart (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) < len (Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) ;

definition
let C be ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ;
let n be ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ;
pred n is_sufficiently_large_for C means :: JORDAN1H:def 3
 ex j being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st
( j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) < width (Gauge (C : ( ( non empty with_non-empty_element ) ( non empty with_non-empty_element ) set ) ,n : ( ( non empty-membered ) ( non empty non empty-membered ) Element of bool K563(C : ( ( non empty with_non-empty_element ) ( non empty with_non-empty_element ) set ) ) : ( ( V218() ) ( V218() ) set ) : ( ( ) ( ) set ) ) )) : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) & cell ((Gauge (C : ( ( non empty with_non-empty_element ) ( non empty with_non-empty_element ) set ) ,n : ( ( non empty-membered ) ( non empty non empty-membered ) Element of bool K563(C : ( ( non empty with_non-empty_element ) ( non empty with_non-empty_element ) set ) ) : ( ( V218() ) ( V218() ) set ) : ( ( ) ( ) set ) ) )) : ( ( tabular ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ,((X-SpanStart (C : ( ( non empty with_non-empty_element ) ( non empty with_non-empty_element ) set ) ,n : ( ( non empty-membered ) ( non empty non empty-membered ) Element of bool K563(C : ( ( non empty with_non-empty_element ) ( non empty with_non-empty_element ) set ) ) : ( ( V218() ) ( V218() ) set ) : ( ( ) ( ) set ) ) )) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) c= BDD C : ( ( non empty with_non-empty_element ) ( non empty with_non-empty_element ) set ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) );
end;

theorem :: JORDAN1H:51
for n being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for C being ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) st n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) is_sufficiently_large_for C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) holds
n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) >= 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ;

theorem :: JORDAN1H:52
for C being ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of )
for n being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) & len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) > 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
for i1, j1 being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st left_cell (f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) meets C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) & [i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. ((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) & [i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) holds
[(i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: JORDAN1H:53
for C being ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of )
for n being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) & len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) > 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
for i1, j1 being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st left_cell (f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) meets C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) & [i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. ((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) & [(i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * ((i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) holds
[(i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: JORDAN1H:54
for C being ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of )
for n being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) & len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) > 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
for j1, i2 being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st left_cell (f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) meets C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) & [(i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. ((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * ((i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) & [i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) holds
[i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: JORDAN1H:55
for C being ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of )
for n being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) & len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) > 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
for i1, j2 being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st left_cell (f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) meets C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) & [i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. ((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) & [i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) holds
[(i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: JORDAN1H:56
for C being ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of )
for n being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) & len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) > 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
for i1, j1 being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st front_left_cell (f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) meets C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) & [i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. ((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) & [i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) holds
[i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: JORDAN1H:57
for C being ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of )
for n being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) & len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) > 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
for i1, j1 being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st front_left_cell (f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) meets C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) & [i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. ((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) & [(i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * ((i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) holds
[(i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: JORDAN1H:58
for C being ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of )
for n being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) & len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) > 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
for j1, i2 being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st front_left_cell (f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) meets C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) & [(i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. ((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * ((i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) & [i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) holds
[(i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: JORDAN1H:59
for C being ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of )
for n being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) & len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) > 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
for i1, j2 being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st front_left_cell (f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) meets C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) & [i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. ((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) & [i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) holds
[i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: JORDAN1H:60
for C being ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of )
for n being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) & len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) > 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
for i1, j1 being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st front_right_cell (f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) meets C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) & [i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. ((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) & [i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) holds
[(i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: JORDAN1H:61
for C being ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of )
for n being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) & len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) > 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
for i1, j1 being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st front_right_cell (f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) meets C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) & [i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. ((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) & [(i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * ((i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) holds
[(i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: JORDAN1H:62
for C being ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of )
for n being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) & len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) > 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
for j1, i2 being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st front_right_cell (f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) meets C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) & [(i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. ((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * ((i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) & [i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) holds
[i2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: JORDAN1H:63
for C being ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of )
for n being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) st f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) is_sequence_on Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) & len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) > 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) holds
for i1, j2 being ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) st front_right_cell (f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ,((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) : ( ( ) ( ) set ) ) meets C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) & [i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. ((len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,(j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) & [i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) /. (len f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() finite-support ) FinSequence of ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) = (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) * (i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V59() V60() V61() finite-support ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) holds
[(i1 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ,j2 : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( non empty compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( ordinal natural V11() real integer finite cardinal ext-real non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) )) : ( ( tabular ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V81() tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence-yielding finite-support ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V11() real integer finite cardinal ext-real positive non negative V46() V69() V70() V71() V72() V73() V74() ) Element of NAT : ( ( ) ( non empty non trivial ordinal non finite cardinal limit_cardinal non empty-membered V69() V70() V71() V72() V73() V74() V75() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered V69() V70() V71() V75() ) set ) : ( ( ) ( non empty non trivial non finite non empty-membered ) set ) ) ) ) : ( ( strict ) ( non empty non trivial TopSpace-like T_2 V156() V181() V182() V183() V184() V185() V186() V187() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty non trivial functional non empty-membered ) set ) ) ) : ( ( ) ( ) set ) ;