:: JGRAPH_1 semantic presentation

begin

theorem :: JGRAPH_1:1
for G being ( ( non empty ) ( non empty V76() ) Graph)
for IT being ( ( oriented ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of b1 : ( ( non empty ) ( non empty V76() ) Graph) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like oriented ) Chain of G : ( ( non empty ) ( non empty V76() ) Graph) )
for vs being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of b1 : ( ( non empty ) ( non empty V76() ) Graph) : ( ( ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of G : ( ( non empty ) ( non empty V76() ) Graph) : ( ( ) ( non empty ) set ) ) st IT : ( ( oriented ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of b1 : ( ( non empty ) ( non empty V76() ) Graph) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like oriented ) Chain of b1 : ( ( non empty ) ( non empty V76() ) Graph) ) is Simple & vs : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of b1 : ( ( non empty ) ( non empty V76() ) Graph) : ( ( ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty V76() ) Graph) : ( ( ) ( non empty ) set ) ) is_oriented_vertex_seq_of IT : ( ( oriented ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of b1 : ( ( non empty ) ( non empty V76() ) Graph) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like oriented ) Chain of b1 : ( ( non empty ) ( non empty V76() ) Graph) ) holds
for n, m being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & n : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) < m : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len vs : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of b1 : ( ( non empty ) ( non empty V76() ) Graph) : ( ( ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty V76() ) Graph) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & vs : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of b1 : ( ( non empty ) ( non empty V76() ) Graph) : ( ( ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty V76() ) Graph) : ( ( ) ( non empty ) set ) ) . n : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = vs : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of b1 : ( ( non empty ) ( non empty V76() ) Graph) : ( ( ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty V76() ) Graph) : ( ( ) ( non empty ) set ) ) . m : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) holds
( n : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) = 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & m : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) = len vs : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of b1 : ( ( non empty ) ( non empty V76() ) Graph) : ( ( ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty ) ( non empty V76() ) Graph) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) ;

definition
let X be ( ( ) ( ) set ) ;
func PGraph X -> ( ( ) ( ) MultiGraphStruct ) equals :: JGRAPH_1:def 1
 MultiGraphStruct(# X : ( ( ) ( ) RLTopStruct ) ,[:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( V1() ) set ) ,(pr1 (X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) )) : ( ( Function-like V18([:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) RLTopStruct ) ) ) ( V1() V4([:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( V1() ) set ) ) V5(X : ( ( ) ( ) RLTopStruct ) ) Function-like V18([:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) RLTopStruct ) ) ) Element of bool [:[:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( ) set ) ) ,(pr2 (X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) )) : ( ( Function-like V18([:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) RLTopStruct ) ) ) ( V1() V4([:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( V1() ) set ) ) V5(X : ( ( ) ( ) RLTopStruct ) ) Function-like V18([:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) RLTopStruct ) ) ) Element of bool [:[:X : ( ( ) ( ) RLTopStruct ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) RLTopStruct ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( ) set ) ) #) : ( ( strict ) ( strict ) MultiGraphStruct ) ;
end;

theorem :: JGRAPH_1:2
for X being ( ( ) ( ) set ) holds the carrier of (PGraph X : ( ( ) ( ) set ) ) : ( ( ) ( ) MultiGraphStruct ) : ( ( ) ( ) set ) = X : ( ( ) ( ) set ) ;

definition
let f be ( ( V1() Function-like FinSequence-like ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence) ;
func PairF f -> ( ( V1() Function-like FinSequence-like ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence) means :: JGRAPH_1:def 2
( len it : ( ( ) ( ) Element of f : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) = (len f : ( ( ) ( ) RLTopStruct ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & ( for i being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) < len f : ( ( ) ( ) RLTopStruct ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) holds
it : ( ( ) ( ) Element of f : ( ( ) ( ) RLTopStruct ) ) . i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = [(f : ( ( ) ( ) RLTopStruct ) . i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) set ) ,(f : ( ( ) ( ) RLTopStruct ) . (i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) ) );
end;

registration
let X be ( ( non empty ) ( non empty ) set ) ;
cluster PGraph X : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) MultiGraphStruct ) -> non empty ;
end;

theorem :: JGRAPH_1:3
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of X : ( ( non empty ) ( non empty ) set ) ) holds f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (PGraph X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( non empty ) set ) ) ;

theorem :: JGRAPH_1:4
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of X : ( ( non empty ) ( non empty ) set ) ) holds PairF f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( V1() Function-like FinSequence-like ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence) is ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier' of (PGraph X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) ;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(X : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of X : ( ( non empty ) ( non empty ) set ) ) ;
:: original: PairF
redefine func PairF f -> ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph X : ( ( ) ( ) set ) ) : ( ( ) ( ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier' of (PGraph X : ( ( ) ( ) set ) ) : ( ( ) ( ) MultiGraphStruct ) : ( ( ) ( ) set ) ) ;
end;

theorem :: JGRAPH_1:5
for X being ( ( non empty ) ( non empty ) set )
for n being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) )
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of X : ( ( non empty ) ( non empty ) set ) ) st 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & n : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len (PairF f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier' of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) holds
(PairF f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier' of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) in the carrier' of (PGraph X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ;

theorem :: JGRAPH_1:6
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of X : ( ( non empty ) ( non empty ) set ) ) holds PairF f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier' of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) is ( ( oriented ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like oriented ) Chain of PGraph X : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) ) ;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(X : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of X : ( ( non empty ) ( non empty ) set ) ) ;
:: original: PairF
redefine func PairF f -> ( ( oriented ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph X : ( ( ) ( ) set ) ) : ( ( ) ( ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like oriented ) Chain of PGraph X : ( ( ) ( ) set ) : ( ( ) ( ) MultiGraphStruct ) ) ;
end;

theorem :: JGRAPH_1:7
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of X : ( ( non empty ) ( non empty ) set ) )
for f1 being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (PGraph X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( non empty ) set ) ) st len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) = f1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( non empty ) set ) ) holds
f1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( non empty ) set ) ) is_oriented_vertex_seq_of PairF f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( oriented ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like oriented ) Chain of PGraph b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) ) ;

begin

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let f, g be ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(X : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of X : ( ( non empty ) ( non empty ) set ) ) ;
pred g is_Shortcut_of f means :: JGRAPH_1:def 3
( f : ( ( ) ( ) set ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = g : ( ( Function-like V18([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) set ) ) ) ( V1() V4([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(X : ( ( ) ( ) set ) ) Function-like V18([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) set ) ) ) Element of bool [:[:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & f : ( ( ) ( ) set ) . (len f : ( ( ) ( ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = g : ( ( Function-like V18([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) set ) ) ) ( V1() V4([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(X : ( ( ) ( ) set ) ) Function-like V18([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) set ) ) ) Element of bool [:[:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( ) set ) ) . (len g : ( ( Function-like V18([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) set ) ) ) ( V1() V4([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(X : ( ( ) ( ) set ) ) Function-like V18([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) set ) ) ) Element of bool [:[:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & ex fc being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph X : ( ( ) ( ) set ) ) : ( ( ) ( ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSubsequence-like ) Subset of ( ( ) ( finite V45() ) set ) ) ex fvs being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(X : ( ( ) ( ) set ) ) Function-like finite FinSubsequence-like ) Subset of ( ( ) ( ) set ) ) ex sc being ( ( oriented simple ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph X : ( ( ) ( ) set ) ) : ( ( ) ( ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like oriented simple ) Chain of PGraph X : ( ( ) ( ) set ) : ( ( ) ( ) MultiGraphStruct ) ) ex g1 being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (PGraph X : ( ( ) ( ) set ) ) : ( ( ) ( ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (PGraph X : ( ( ) ( ) set ) ) : ( ( ) ( ) MultiGraphStruct ) : ( ( ) ( ) set ) ) st
( Seq fc : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSubsequence-like ) Subset of ( ( ) ( finite V45() ) set ) ) : ( ( V1() Function-like ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite FinSequence-like FinSubsequence-like ) set ) = sc : ( ( oriented simple ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like oriented simple ) Chain of PGraph X : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) ) & Seq fvs : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(X : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSubsequence-like ) Subset of ( ( ) ( finite V45() ) set ) ) : ( ( V1() Function-like ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite FinSequence-like FinSubsequence-like ) set ) = g : ( ( Function-like V18([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) set ) ) ) ( V1() V4([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(X : ( ( ) ( ) set ) ) Function-like V18([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) set ) ) ) Element of bool [:[:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( ) set ) ) & g1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (PGraph X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (PGraph X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( non empty ) set ) ) = g : ( ( Function-like V18([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) set ) ) ) ( V1() V4([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ) V5(X : ( ( ) ( ) set ) ) Function-like V18([:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) set ) ) ) Element of bool [:[:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( V1() ) set ) : ( ( ) ( ) set ) ) & g1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (PGraph X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of the carrier of (PGraph X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( non empty ) set ) ) is_oriented_vertex_seq_of sc : ( ( oriented simple ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like oriented simple ) Chain of PGraph X : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) ) ) );
end;

theorem :: JGRAPH_1:8
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of X : ( ( non empty ) ( non empty ) set ) ) st g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is_Shortcut_of f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) holds
( 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & len g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: JGRAPH_1:9
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of X : ( ( non empty ) ( non empty ) set ) ) st len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) holds
ex g being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of X : ( ( non empty ) ( non empty ) set ) ) st g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is_Shortcut_of f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: JGRAPH_1:10
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of X : ( ( non empty ) ( non empty ) set ) ) st g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is_Shortcut_of f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) holds
rng (PairF g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( oriented ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like oriented ) Chain of PGraph b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) ) : ( ( ) ( finite ) Element of bool the carrier' of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) c= rng (PairF f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( oriented ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like oriented ) Chain of PGraph b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) ) : ( ( ) ( finite ) Element of bool the carrier' of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_1:11
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of X : ( ( non empty ) ( non empty ) set ) ) st f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) <> f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) . (len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is_Shortcut_of f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) holds
g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is one-to-one ;

definition
let n be ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ;
let IT be ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL n : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ;
attr IT is nodic means :: JGRAPH_1:def 4
 for i, j being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) holds
( not LSeg (IT : ( ( ) ( ) set ) ,i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL n : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) meets LSeg (IT : ( ( ) ( ) set ) ,j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL n : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) or ( (LSeg (IT : ( ( ) ( ) set ) ,i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL n : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) /\ (LSeg (IT : ( ( ) ( ) set ) ,j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL n : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL n : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) = {(IT : ( ( ) ( ) set ) . i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) set ) } : ( ( ) ( non empty trivial finite 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element ) set ) & ( IT : ( ( ) ( ) set ) . i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = IT : ( ( ) ( ) set ) . j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) or IT : ( ( ) ( ) set ) . i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = IT : ( ( ) ( ) set ) . (j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) or ( (LSeg (IT : ( ( ) ( ) set ) ,i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL n : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) /\ (LSeg (IT : ( ( ) ( ) set ) ,j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL n : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool the carrier of (TOP-REAL n : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) = {(IT : ( ( ) ( ) set ) . (i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) set ) } : ( ( ) ( non empty trivial finite 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element ) set ) & ( IT : ( ( ) ( ) set ) . (i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = IT : ( ( ) ( ) set ) . j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) or IT : ( ( ) ( ) set ) . (i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = IT : ( ( ) ( ) set ) . (j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) or LSeg (IT : ( ( ) ( ) set ) ,i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL n : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) = LSeg (IT : ( ( ) ( ) set ) ,j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL n : ( ( ) ( ) set ) ) : ( ( strict ) ( strict ) RLTopStruct ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) );
end;

theorem :: JGRAPH_1:12
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) st f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is s.n.c. holds
f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is s.c.c. ;

theorem :: JGRAPH_1:13
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) st f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is s.c.c. & LSeg (f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ,1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) misses LSeg (f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ,((len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) holds
f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is s.n.c. ;

theorem :: JGRAPH_1:14
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) st f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is nodic & PairF f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( oriented ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like oriented ) Chain of PGraph the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) ) is Simple holds
f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is s.c.c. ;

theorem :: JGRAPH_1:15
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) st f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is nodic & PairF f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( oriented ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like oriented ) Chain of PGraph the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) ) is Simple & f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) <> f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . (len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) holds
f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is s.n.c. ;

theorem :: JGRAPH_1:16
for n being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) )
for p1, p2, p3 being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) holds
( for x being ( ( ) ( ) set ) holds
( not x : ( ( ) ( ) set ) <> p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) or not x : ( ( ) ( ) set ) in (LSeg (p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ,p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) )) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) /\ (LSeg (p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ,p3 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) )) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) ) or p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) in LSeg (p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ,p3 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) or p3 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) in LSeg (p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ,p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: JGRAPH_1:17
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) st f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is s.n.c. & (LSeg (f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ,1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) /\ (LSeg (f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ,(1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) c= {(f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. (1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) } : ( ( ) ( functional non empty trivial finite V45() 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element ) set ) & (LSeg (f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ,((len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -' 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) /\ (LSeg (f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ,((len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) c= {(f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. ((len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) } : ( ( ) ( functional non empty trivial finite V45() 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element ) set ) holds
f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is unfolded ;

theorem :: JGRAPH_1:18
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of X : ( ( non empty ) ( non empty ) set ) ) st PairF f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( oriented ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like oriented ) Chain of PGraph b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) ) is Simple & f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) <> f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) . (len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) holds
( f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is one-to-one & len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <> 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: JGRAPH_1:19
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of X : ( ( non empty ) ( non empty ) set ) ) st f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) is one-to-one & len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) > 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) holds
( PairF f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( oriented ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like oriented ) Chain of PGraph b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) ) is Simple & f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) <> f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) . (len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5(b1 : ( ( non empty ) ( non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_1:20
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) st f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is nodic & PairF f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( oriented ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier' of (PGraph the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) : ( ( ) ( ) set ) ) Function-like finite FinSequence-like FinSubsequence-like oriented ) Chain of PGraph the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( non empty V76() ) MultiGraphStruct ) ) is Simple & f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) <> f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . (len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) holds
f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is unfolded ;

theorem :: JGRAPH_1:21
for f, g being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) )
for i being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) st g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is_Shortcut_of f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) & 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) holds
ex k1 being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= k1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & k1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. k1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) = g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) & f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. (k1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) = g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. (i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) & f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . k1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) = g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) & f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . (k1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) = g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . (i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) ) ;

theorem :: JGRAPH_1:22
for f, g being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) st g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is_Shortcut_of f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) holds
rng g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( functional finite ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) c= rng f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( functional finite ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_1:23
for f, g being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) st g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is_Shortcut_of f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) holds
L~ g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) c= L~ f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) ;

theorem :: JGRAPH_1:24
for f, g being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) st f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is special & g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is_Shortcut_of f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) holds
g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is special ;

theorem :: JGRAPH_1:25
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) st f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is special & 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) <> f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . (len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) holds
ex g being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) st
( 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is special & g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is one-to-one & L~ g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) c= L~ f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) & f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) = g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) & f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . (len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) = g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . (len g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) & rng g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( functional finite ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) c= rng f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( functional finite ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: JGRAPH_1:26
for f1, f2 being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) st f1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is special & f2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is special & 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len f1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len f2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & f1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) <> f1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . (len f1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) & f2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) <> f2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . (len f2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) & X_axis f1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) lies_between (X_axis f1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ,(X_axis f1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . (len f1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & X_axis f2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) lies_between (X_axis f1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ,(X_axis f1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . (len f1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & Y_axis f1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) lies_between (Y_axis f2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ,(Y_axis f2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . (len f2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & Y_axis f2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) lies_between (Y_axis f2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ,(Y_axis f2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . (len f2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) holds
L~ f1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) meets L~ f2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) ;

begin

theorem :: JGRAPH_1:27
for a, b, r1, r2 being ( ( ) ( V28() real ext-real ) Real) st a : ( ( ) ( V28() real ext-real ) Real) <= r1 : ( ( ) ( V28() real ext-real ) Real) & r1 : ( ( ) ( V28() real ext-real ) Real) <= b : ( ( ) ( V28() real ext-real ) Real) & a : ( ( ) ( V28() real ext-real ) Real) <= r2 : ( ( ) ( V28() real ext-real ) Real) & r2 : ( ( ) ( V28() real ext-real ) Real) <= b : ( ( ) ( V28() real ext-real ) Real) holds
abs (r1 : ( ( ) ( V28() real ext-real ) Real) - r2 : ( ( ) ( V28() real ext-real ) Real) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= b : ( ( ) ( V28() real ext-real ) Real) - a : ( ( ) ( V28() real ext-real ) Real) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ;

theorem :: JGRAPH_1:28
for n being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) )
for p1, p2 being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) )
for x1, x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) holds
|.(p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) - p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) = dist (x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ;

theorem :: JGRAPH_1:29
for p being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) holds |.p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ^2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) = ((p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `1) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ^2) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) + ((p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `2) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ^2) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ;

theorem :: JGRAPH_1:30
for p being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) holds |.p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) = sqrt (((p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `1) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ^2) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) + ((p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `2) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ^2) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ;

theorem :: JGRAPH_1:31
for p being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) holds |.p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= (abs (p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `1) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) + (abs (p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `2) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ;

theorem :: JGRAPH_1:32
for p1, p2 being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) holds |.(p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) - p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= (abs ((p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `1) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) - (p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `1) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) + (abs ((p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `2) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) - (p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `2) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ;

theorem :: JGRAPH_1:33
for p being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) holds
( abs (p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `1) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= |.p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & abs (p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `2) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= |.p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) ;

theorem :: JGRAPH_1:34
for p1, p2 being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) holds
( abs ((p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `1) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) - (p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `1) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= |.(p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) - p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & abs ((p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `2) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) - (p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `2) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= |.(p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) - p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) ;

theorem :: JGRAPH_1:35
for n being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) )
for p, p1, p2 being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) st p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) in LSeg (p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ,p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) holds
ex r being ( ( ) ( V28() real ext-real ) Real) st
( 0 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= r : ( ( ) ( V28() real ext-real ) Real) & r : ( ( ) ( V28() real ext-real ) Real) <= 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) = ((1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) - r : ( ( ) ( V28() real ext-real ) Real) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) * p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) + (r : ( ( ) ( V28() real ext-real ) Real) * p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ;

theorem :: JGRAPH_1:36
for n being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) )
for p, p1, p2 being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) st p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) in LSeg (p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ,p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( functional ) Element of bool the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) holds
( |.(p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) - p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= |.(p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) - p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & |.(p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) - p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= |.(p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) - p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) ;

begin

theorem :: JGRAPH_1:37
for M being ( ( non empty Reflexive discerning V105() triangle ) ( non empty Reflexive discerning V105() triangle Discerning ) MetrSpace)
for P, Q being ( ( ) ( ) Subset of ) st P : ( ( ) ( ) Subset of ) <> {} : ( ( ) ( ) set ) & P : ( ( ) ( ) Subset of ) is compact & Q : ( ( ) ( ) Subset of ) <> {} : ( ( ) ( ) set ) & Q : ( ( ) ( ) Subset of ) is compact holds
min_dist_min (P : ( ( ) ( ) Subset of ) ,Q : ( ( ) ( ) Subset of ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) >= 0 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: JGRAPH_1:38
for M being ( ( non empty Reflexive discerning V105() triangle ) ( non empty Reflexive discerning V105() triangle Discerning ) MetrSpace)
for P, Q being ( ( ) ( ) Subset of ) st P : ( ( ) ( ) Subset of ) <> {} : ( ( ) ( ) set ) & P : ( ( ) ( ) Subset of ) is compact & Q : ( ( ) ( ) Subset of ) <> {} : ( ( ) ( ) set ) & Q : ( ( ) ( ) Subset of ) is compact holds
( P : ( ( ) ( ) Subset of ) misses Q : ( ( ) ( ) Subset of ) iff min_dist_min (P : ( ( ) ( ) Subset of ) ,Q : ( ( ) ( ) Subset of ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) > 0 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: JGRAPH_1:39
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) )
for a, c, d being ( ( ) ( V28() real ext-real ) Real) st 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & X_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) lies_between (X_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ,(X_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . (len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & Y_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) lies_between c : ( ( ) ( V28() real ext-real ) Real) ,d : ( ( ) ( V28() real ext-real ) Real) & a : ( ( ) ( V28() real ext-real ) Real) > 0 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & ( for i being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) & i : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) holds
|.((f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. i : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) - (f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. (i : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) < a : ( ( ) ( V28() real ext-real ) Real) ) holds
ex g being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) st
( g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is special & g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) = f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) & g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . (len g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) = f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . (len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) & len g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) >= len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & X_axis g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) lies_between (X_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ,(X_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . (len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & Y_axis g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) lies_between c : ( ( ) ( V28() real ext-real ) Real) ,d : ( ( ) ( V28() real ext-real ) Real) & ( for j being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) st j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) in dom g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( finite V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of bool NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( non trivial non finite ) set ) ) holds
ex k being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) st
( k : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) in dom f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( finite V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of bool NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( non trivial non finite ) set ) ) & |.((g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) - (f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. k : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) < a : ( ( ) ( V28() real ext-real ) Real) ) ) & ( for j being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) holds
|.((g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) - (g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. (j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) < a : ( ( ) ( V28() real ext-real ) Real) ) ) ;

theorem :: JGRAPH_1:40
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) )
for a, c, d being ( ( ) ( V28() real ext-real ) Real) st 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & Y_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) lies_between (Y_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ,(Y_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . (len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & X_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) lies_between c : ( ( ) ( V28() real ext-real ) Real) ,d : ( ( ) ( V28() real ext-real ) Real) & a : ( ( ) ( V28() real ext-real ) Real) > 0 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & ( for i being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) & i : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) holds
|.((f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. i : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) - (f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. (i : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) < a : ( ( ) ( V28() real ext-real ) Real) ) holds
ex g being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) st
( g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) is special & g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) = f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) & g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . (len g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) = f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) . (len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() Function-like ) set ) & len g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) >= len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & Y_axis g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) lies_between (Y_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ,(Y_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . (len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & X_axis g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) lies_between c : ( ( ) ( V28() real ext-real ) Real) ,d : ( ( ) ( V28() real ext-real ) Real) & ( for j being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) st j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) in dom g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( finite V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of bool NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( non trivial non finite ) set ) ) holds
ex k being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) st
( k : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) in dom f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( finite V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of bool NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( non trivial non finite ) set ) ) & |.((g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) - (f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. k : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) < a : ( ( ) ( V28() real ext-real ) Real) ) ) & ( for j being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) holds
|.((g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) - (g : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. (j : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) .| : ( ( ) ( V28() real ext-real non negative ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) < a : ( ( ) ( V28() real ext-real ) Real) ) ) ;

theorem :: JGRAPH_1:41
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) st 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) holds
( len (X_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) = len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & (X_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) = (f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `1 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & (X_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . (len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) = (f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. (len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `1 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) ;

theorem :: JGRAPH_1:42
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) st 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) <= len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) holds
( len (Y_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) = len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & (Y_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) = (f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & (Y_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . (len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) = (f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. (len f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) ;

theorem :: JGRAPH_1:43
for i being ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) )
for f being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) st i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) in dom f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) : ( ( ) ( finite V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of bool NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( non trivial non finite ) set ) ) holds
( (X_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) = (f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `1 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & (Y_axis f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) Function-like finite FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) FinSequence of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) . i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) = (f : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like finite FinSequence-like FinSubsequence-like ) FinSequence of ( ( ) ( functional non empty ) set ) ) /. i : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) ;

theorem :: JGRAPH_1:44
for P, Q being ( ( non empty ) ( functional non empty ) Subset of )
for p1, p2, q1, q2 being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) st P : ( ( non empty ) ( functional non empty ) Subset of ) is_an_arc_of p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ,p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) & Q : ( ( non empty ) ( functional non empty ) Subset of ) is_an_arc_of q1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) ,q2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) & ( for p being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) st p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) in P : ( ( non empty ) ( functional non empty ) Subset of ) holds
( p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `1 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `1 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `1 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `1 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) ) & ( for p being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) st p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) in Q : ( ( non empty ) ( functional non empty ) Subset of ) holds
( p1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `1 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `1 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `1 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= p2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `1 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) ) & ( for p being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) st p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) in P : ( ( non empty ) ( functional non empty ) Subset of ) holds
( q1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= q2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) ) & ( for p being ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) st p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) in Q : ( ( non empty ) ( functional non empty ) Subset of ) holds
( q1 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & p : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= q2 : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Point of ( ( ) ( functional non empty ) set ) ) `2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) ) ) holds
P : ( ( non empty ) ( functional non empty ) Subset of ) meets Q : ( ( non empty ) ( functional non empty ) Subset of ) ;

theorem :: JGRAPH_1:45
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V5( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for P being ( ( non empty ) ( non empty ) Subset of )
for f1 being ( ( Function-like V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V5( the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) Function-like non empty total V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st f : ( ( Function-like V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V5( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f1 : ( ( Function-like V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V5( the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) Function-like non empty total V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & f : ( ( Function-like V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V5( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous holds
f1 : ( ( Function-like V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V5( the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) Function-like non empty total V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous ;

theorem :: JGRAPH_1:46
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V5( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for P being ( ( non empty ) ( non empty ) Subset of ) st X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) is compact & Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) is T_2 & f : ( ( Function-like V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V5( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V5( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & P : ( ( non empty ) ( non empty ) Subset of ) = rng f : ( ( Function-like V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V5( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) ) holds
ex f1 being ( ( Function-like V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V5( the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) Function-like non empty total V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st
( f : ( ( Function-like V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V5( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) Function-like non empty total V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f1 : ( ( Function-like V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V5( the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) Function-like non empty total V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & f1 : ( ( Function-like V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) ) ( V1() V4( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V5( the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) Function-like non empty total V18( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b4 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is being_homeomorphism ) ;

theorem :: JGRAPH_1:47
for f, g being ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) )
for a, b, c, d being ( ( real ) ( V28() real ext-real ) number )
for O, I being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = 0 : ( ( ) ( ordinal natural V28() real ext-real non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = 1 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) & f : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) is continuous & f : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) is one-to-one & g : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) is continuous & g : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) is one-to-one & (f : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `1 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) = a : ( ( real ) ( V28() real ext-real ) number ) & (f : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `1 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) = b : ( ( real ) ( V28() real ext-real ) number ) & (g : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) . O : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) = c : ( ( real ) ( V28() real ext-real ) number ) & (g : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) . I : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) = d : ( ( real ) ( V28() real ext-real ) number ) & ( for r being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( a : ( ( real ) ( V28() real ext-real ) number ) <= (f : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) . r : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `1 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & (f : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) . r : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `1 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= b : ( ( real ) ( V28() real ext-real ) number ) & a : ( ( real ) ( V28() real ext-real ) number ) <= (g : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) . r : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `1 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & (g : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) . r : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `1 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= b : ( ( real ) ( V28() real ext-real ) number ) & c : ( ( real ) ( V28() real ext-real ) number ) <= (f : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) . r : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & (f : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) . r : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= d : ( ( real ) ( V28() real ext-real ) number ) & c : ( ( real ) ( V28() real ext-real ) number ) <= (g : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) . r : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) & (g : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) . r : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V1() V4( NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) Function-like finite 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-yielding V172() V173() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) `2 : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) ) <= d : ( ( real ) ( V28() real ext-real ) number ) ) ) holds
rng f : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) : ( ( ) ( functional non empty ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) meets rng g : ( ( Function-like V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) ( V1() V4( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V5( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) Function-like non empty total V18( the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like ) SubSpace of K716() : ( ( TopSpace-like ) ( TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( functional non empty ) set ) ) : ( ( ) ( functional non empty ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V28() real ext-real positive non negative V33() V38() finite cardinal V181() V182() V183() V184() V185() V186() V281() V282() V283() V284() V285() ) Element of NAT : ( ( ) ( non trivial ordinal non finite cardinal limit_cardinal V181() V182() V183() V184() V185() V186() V187() V283() ) Element of bool REAL : ( ( ) ( V181() V182() V183() V187() V283() V284() V286() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V128() V194() V211() V212() V213() V214() V215() V216() V217() strict ) RLTopStruct ) : ( ( ) ( functional non empty ) set ) : ( ( ) ( ) set ) ) ;