begin
definition
let X be ( ( ) ( )
set ) ;
let CNS1,
CNS2 be ( ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like ) ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like )
ComplexNormSpace) ;
let f be ( (
Function-like ) (
Relation-like the
carrier of
CNS1 : ( ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like ) ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) ( non
empty )
set )
-defined the
carrier of
CNS2 : ( ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like ) ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) ( non
empty )
set )
-valued Function-like )
PartFunc of ,) ;
pred f is_uniformly_continuous_on X means
(
X : ( ( ) ( )
CNORMSTR )
c= dom f : ( (
Function-like quasi_total ) (
Relation-like K7(
COMPLEX : ( ( ) ( non
empty V47()
V53()
V54() )
set ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set )
-defined X : ( ( ) ( )
CNORMSTR )
-valued Function-like quasi_total )
Element of
K6(
K7(
K7(
COMPLEX : ( ( ) ( non
empty V47()
V53()
V54() )
set ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
Relation-like )
Element of
K6( the
carrier of
CNS1 : ( ( ) ( )
Element of
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) & ( for
r being ( ( ) (
V11()
V12()
ext-real )
Real) st
0 : ( ( ) (
empty V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative Function-like functional V45()
V46()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K6(
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V11()
V12()
ext-real )
Real) holds
ex
s being ( ( ) (
V11()
V12()
ext-real )
Real) st
(
0 : ( ( ) (
empty V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative Function-like functional V45()
V46()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K6(
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
< s : ( ( ) (
V11()
V12()
ext-real )
Real) & ( for
x1,
x2 being ( ( ) ( )
Point of ( ( ) ( )
set ) ) st
x1 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) )
in X : ( ( ) ( )
CNORMSTR ) &
x2 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) )
in X : ( ( ) ( )
CNORMSTR ) &
||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS1 : ( ( ) ( ) Element of X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) .|| : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) )
< s : ( ( ) (
V11()
V12()
ext-real )
Real) holds
||.((f : ( ( Function-like quasi_total ) ( Relation-like K7(COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS2 : ( ( Function-like quasi_total ) ( Relation-like K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (f : ( ( Function-like quasi_total ) ( Relation-like K7(COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS2 : ( ( Function-like quasi_total ) ( Relation-like K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS2 : ( ( Function-like quasi_total ) ( Relation-like K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) )
< r : ( ( ) (
V11()
V12()
ext-real )
Real) ) ) ) );
end;
definition
let X be ( ( ) ( )
set ) ;
let RNS be ( ( non
empty V107()
V132()
V133()
V134()
V135()
V136()
V137()
V138()
V142()
V143()
RealNormSpace-like ) ( non
empty V107()
V132()
V133()
V134()
V135()
V136()
V137()
V138()
V142()
V143()
RealNormSpace-like )
RealNormSpace) ;
let CNS be ( ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like ) ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like )
ComplexNormSpace) ;
let f be ( (
Function-like ) (
Relation-like the
carrier of
CNS : ( ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like ) ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) ( non
empty )
set )
-defined the
carrier of
RNS : ( ( non
empty V107()
V132()
V133()
V134()
V135()
V136()
V137()
V138()
V142()
V143()
RealNormSpace-like ) ( non
empty V107()
V132()
V133()
V134()
V135()
V136()
V137()
V138()
V142()
V143()
RealNormSpace-like )
RealNormSpace) : ( ( ) ( non
empty )
set )
-valued Function-like )
PartFunc of ,) ;
pred f is_uniformly_continuous_on X means
(
X : ( ( ) ( )
CNORMSTR )
c= dom f : ( (
Function-like quasi_total ) (
Relation-like K7(
COMPLEX : ( ( ) ( non
empty V47()
V53()
V54() )
set ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set )
-defined X : ( ( ) ( )
CNORMSTR )
-valued Function-like quasi_total )
Element of
K6(
K7(
K7(
COMPLEX : ( ( ) ( non
empty V47()
V53()
V54() )
set ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
Relation-like )
Element of
K6( the
carrier of
CNS : ( (
Function-like quasi_total ) (
Relation-like K7(
X : ( ( ) ( )
CNORMSTR ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set )
-defined X : ( ( ) ( )
CNORMSTR )
-valued Function-like quasi_total )
Element of
K6(
K7(
K7(
X : ( ( ) ( )
CNORMSTR ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) & ( for
r being ( ( ) (
V11()
V12()
ext-real )
Real) st
0 : ( ( ) (
empty V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative Function-like functional V45()
V46()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K6(
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V11()
V12()
ext-real )
Real) holds
ex
s being ( ( ) (
V11()
V12()
ext-real )
Real) st
(
0 : ( ( ) (
empty V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative Function-like functional V45()
V46()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K6(
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
< s : ( ( ) (
V11()
V12()
ext-real )
Real) & ( for
x1,
x2 being ( ( ) ( )
Point of ( ( ) ( )
set ) ) st
x1 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) )
in X : ( ( ) ( )
CNORMSTR ) &
x2 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) )
in X : ( ( ) ( )
CNORMSTR ) &
||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( Function-like quasi_total ) ( Relation-like K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) )
< s : ( ( ) (
V11()
V12()
ext-real )
Real) holds
||.((f : ( ( Function-like quasi_total ) ( Relation-like K7(COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) Element of X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) - (f : ( ( Function-like quasi_total ) ( Relation-like K7(COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) Element of X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) Element of X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) .|| : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) )
< r : ( ( ) (
V11()
V12()
ext-real )
Real) ) ) ) );
end;
definition
let X be ( ( ) ( )
set ) ;
let RNS be ( ( non
empty V107()
V132()
V133()
V134()
V135()
V136()
V137()
V138()
V142()
V143()
RealNormSpace-like ) ( non
empty V107()
V132()
V133()
V134()
V135()
V136()
V137()
V138()
V142()
V143()
RealNormSpace-like )
RealNormSpace) ;
let CNS be ( ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like ) ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like )
ComplexNormSpace) ;
let f be ( (
Function-like ) (
Relation-like the
carrier of
RNS : ( ( non
empty V107()
V132()
V133()
V134()
V135()
V136()
V137()
V138()
V142()
V143()
RealNormSpace-like ) ( non
empty V107()
V132()
V133()
V134()
V135()
V136()
V137()
V138()
V142()
V143()
RealNormSpace-like )
RealNormSpace) : ( ( ) ( non
empty )
set )
-defined the
carrier of
CNS : ( ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like ) ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) ( non
empty )
set )
-valued Function-like )
PartFunc of ,) ;
pred f is_uniformly_continuous_on X means
(
X : ( ( ) ( )
CNORMSTR )
c= dom f : ( (
Function-like quasi_total ) (
Relation-like K7(
COMPLEX : ( ( ) ( non
empty V47()
V53()
V54() )
set ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set )
-defined X : ( ( ) ( )
CNORMSTR )
-valued Function-like quasi_total )
Element of
K6(
K7(
K7(
COMPLEX : ( ( ) ( non
empty V47()
V53()
V54() )
set ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
Relation-like )
Element of
K6( the
carrier of
RNS : ( ( ) ( )
Element of
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) & ( for
r being ( ( ) (
V11()
V12()
ext-real )
Real) st
0 : ( ( ) (
empty V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative Function-like functional V45()
V46()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K6(
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V11()
V12()
ext-real )
Real) holds
ex
s being ( ( ) (
V11()
V12()
ext-real )
Real) st
(
0 : ( ( ) (
empty V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative Function-like functional V45()
V46()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K6(
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
< s : ( ( ) (
V11()
V12()
ext-real )
Real) & ( for
x1,
x2 being ( ( ) ( )
Point of ( ( ) ( )
set ) ) st
x1 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) )
in X : ( ( ) ( )
CNORMSTR ) &
x2 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) )
in X : ( ( ) ( )
CNORMSTR ) &
||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) Element of X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) .|| : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) )
< s : ( ( ) (
V11()
V12()
ext-real )
Real) holds
||.((f : ( ( Function-like quasi_total ) ( Relation-like K7(COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( Function-like quasi_total ) ( Relation-like K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (f : ( ( Function-like quasi_total ) ( Relation-like K7(COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( Function-like quasi_total ) ( Relation-like K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( Function-like quasi_total ) ( Relation-like K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) )
< r : ( ( ) (
V11()
V12()
ext-real )
Real) ) ) ) );
end;
definition
let X be ( ( ) ( )
set ) ;
let CNS be ( ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like ) ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like )
ComplexNormSpace) ;
let f be ( (
Function-like ) (
Relation-like the
carrier of
CNS : ( ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like ) ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V47()
V53()
V54() )
set )
-valued Function-like complex-valued )
PartFunc of ,) ;
pred f is_uniformly_continuous_on X means
(
X : ( ( ) ( )
CNORMSTR )
c= dom f : ( (
Function-like quasi_total ) (
Relation-like K7(
X : ( ( ) ( )
CNORMSTR ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set )
-defined X : ( ( ) ( )
CNORMSTR )
-valued Function-like quasi_total )
Element of
K6(
K7(
K7(
X : ( ( ) ( )
CNORMSTR ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
Relation-like )
Element of
K6( the
carrier of
CNS : ( ( ) ( )
Element of
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) & ( for
r being ( ( ) (
V11()
V12()
ext-real )
Real) st
0 : ( ( ) (
empty V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative Function-like functional V45()
V46()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K6(
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V11()
V12()
ext-real )
Real) holds
ex
s being ( ( ) (
V11()
V12()
ext-real )
Real) st
(
0 : ( ( ) (
empty V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative Function-like functional V45()
V46()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K6(
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
< s : ( ( ) (
V11()
V12()
ext-real )
Real) & ( for
x1,
x2 being ( ( ) ( )
Point of ( ( ) ( )
set ) ) st
x1 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) )
in X : ( ( ) ( )
CNORMSTR ) &
x2 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) )
in X : ( ( ) ( )
CNORMSTR ) &
||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( ) ( ) Element of X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) .|| : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) )
< s : ( ( ) (
V11()
V12()
ext-real )
Real) holds
|.((f : ( ( Function-like quasi_total ) ( Relation-like K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ) - (f : ( ( Function-like quasi_total ) ( Relation-like K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ) .| : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) )
< r : ( ( ) (
V11()
V12()
ext-real )
Real) ) ) ) );
end;
definition
let X be ( ( ) ( )
set ) ;
let CNS be ( ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like ) ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like )
ComplexNormSpace) ;
let f be ( (
Function-like ) (
Relation-like the
carrier of
CNS : ( ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like ) ( non
empty V107()
V132()
V133()
V134()
V142()
V143()
V148()
V149()
V150()
V151()
ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) ( non
empty )
set )
-defined REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set )
-valued Function-like complex-valued ext-real-valued real-valued )
PartFunc of ,) ;
pred f is_uniformly_continuous_on X means
(
X : ( ( ) ( )
CNORMSTR )
c= dom f : ( (
Function-like quasi_total ) (
Relation-like K7(
X : ( ( ) ( )
CNORMSTR ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set )
-defined X : ( ( ) ( )
CNORMSTR )
-valued Function-like quasi_total )
Element of
K6(
K7(
K7(
X : ( ( ) ( )
CNORMSTR ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
Relation-like )
Element of
K6( the
carrier of
CNS : ( ( ) ( )
Element of
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) & ( for
r being ( ( ) (
V11()
V12()
ext-real )
Real) st
0 : ( ( ) (
empty V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative Function-like functional V45()
V46()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K6(
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V11()
V12()
ext-real )
Real) holds
ex
s being ( ( ) (
V11()
V12()
ext-real )
Real) st
(
0 : ( ( ) (
empty V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative Function-like functional V45()
V46()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K6(
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
< s : ( ( ) (
V11()
V12()
ext-real )
Real) & ( for
x1,
x2 being ( ( ) ( )
Point of ( ( ) ( )
set ) ) st
x1 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) )
in X : ( ( ) ( )
CNORMSTR ) &
x2 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) )
in X : ( ( ) ( )
CNORMSTR ) &
||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of CNS : ( ( ) ( ) Element of X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) .|| : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) )
< s : ( ( ) (
V11()
V12()
ext-real )
Real) holds
abs ((f : ( ( Function-like quasi_total ) ( Relation-like K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V47() V48() V49() V53() V54() ) set ) ) - (f : ( ( Function-like quasi_total ) ( Relation-like K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V47() V48() V49() V53() V54() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) )
< r : ( ( ) (
V11()
V12()
ext-real )
Real) ) ) ) );
end;
definition
let X be ( ( ) ( )
set ) ;
let RNS be ( ( non
empty V107()
V132()
V133()
V134()
V135()
V136()
V137()
V138()
V142()
V143()
RealNormSpace-like ) ( non
empty V107()
V132()
V133()
V134()
V135()
V136()
V137()
V138()
V142()
V143()
RealNormSpace-like )
RealNormSpace) ;
let f be ( (
Function-like ) (
Relation-like the
carrier of
RNS : ( ( non
empty V107()
V132()
V133()
V134()
V135()
V136()
V137()
V138()
V142()
V143()
RealNormSpace-like ) ( non
empty V107()
V132()
V133()
V134()
V135()
V136()
V137()
V138()
V142()
V143()
RealNormSpace-like )
RealNormSpace) : ( ( ) ( non
empty )
set )
-defined COMPLEX : ( ( ) ( non
empty V47()
V53()
V54() )
set )
-valued Function-like complex-valued )
PartFunc of ,) ;
pred f is_uniformly_continuous_on X means
(
X : ( ( ) ( )
CNORMSTR )
c= dom f : ( (
Function-like quasi_total ) (
Relation-like K7(
X : ( ( ) ( )
CNORMSTR ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set )
-defined X : ( ( ) ( )
CNORMSTR )
-valued Function-like quasi_total )
Element of
K6(
K7(
K7(
X : ( ( ) ( )
CNORMSTR ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
Relation-like )
Element of
K6( the
carrier of
RNS : ( ( ) ( )
Element of
X : ( ( ) ( )
CNORMSTR ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) & ( for
r being ( ( ) (
V11()
V12()
ext-real )
Real) st
0 : ( ( ) (
empty V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative Function-like functional V45()
V46()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K6(
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V11()
V12()
ext-real )
Real) holds
ex
s being ( ( ) (
V11()
V12()
ext-real )
Real) st
(
0 : ( ( ) (
empty V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative Function-like functional V45()
V46()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K6(
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
< s : ( ( ) (
V11()
V12()
ext-real )
Real) & ( for
x1,
x2 being ( ( ) ( )
Point of ( ( ) ( )
set ) ) st
x1 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) )
in X : ( ( ) ( )
CNORMSTR ) &
x2 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) )
in X : ( ( ) ( )
CNORMSTR ) &
||.(x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of RNS : ( ( ) ( ) Element of X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) .|| : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) )
< s : ( ( ) (
V11()
V12()
ext-real )
Real) holds
|.((f : ( ( Function-like quasi_total ) ( Relation-like K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ) - (f : ( ( Function-like quasi_total ) ( Relation-like K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) CNORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(X : ( ( ) ( ) CNORMSTR ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) CNORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non empty V47() V53() V54() ) set ) ) .| : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V47()
V48()
V49()
V53()
V54() )
set ) )
< r : ( ( ) (
V11()
V12()
ext-real )
Real) ) ) ) );
end;
begin