begin
theorem
for
X,
Y being ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace)
for
f being ( (
Function-like V33( the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) , the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set )
-valued Function-like V33( the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) , the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) )
Function of ( ( ) ( non
empty )
set ) , ( ( ) ( non
empty )
set ) )
for
S being ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
b1 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) holds
f : ( (
Function-like V33( the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) , the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set )
-valued Function-like V33( the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) , the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) )
Function of ( ( ) ( non
empty )
set ) , ( ( ) ( non
empty )
set ) )
* S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
b1 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set )
-valued Function-like )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) is ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
b2 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) ;
theorem
for
X,
Y being ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace)
for
f being ( (
Function-like V33( the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) , the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set )
-valued Function-like V33( the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) , the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) )
Function of ( ( ) ( non
empty )
set ) , ( ( ) ( non
empty )
set ) )
for
S being ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
b1 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) )
for
T being ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
b2 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) st
S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
b1 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) is
convergent &
T : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
b2 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) )
= f : ( (
Function-like V33( the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) , the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set )
-valued Function-like V33( the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) , the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) )
Function of ( ( ) ( non
empty )
set ) , ( ( ) ( non
empty )
set ) )
* S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
b1 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set )
-valued Function-like )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
Function-like V33( the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) , the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set )
-defined the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set )
-valued Function-like V33( the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) , the
carrier of
(TopSpaceMetr b2 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) )
Function of ( ( ) ( non
empty )
set ) , ( ( ) ( non
empty )
set ) ) is
continuous holds
T : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
b2 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) is
convergent ;
theorem
for
s being ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence)
for
S being ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) )
V158()
V159()
V160() )
sequence of ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) st
s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence)
= S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) )
V158()
V159()
V160() )
sequence of ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) holds
( (
s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) is
convergent implies
S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) )
V158()
V159()
V160() )
sequence of ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) is
convergent ) & (
S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) )
V158()
V159()
V160() )
sequence of ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) is
convergent implies
s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) is
convergent ) & (
s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) is
convergent implies
lim s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
= lim S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) )
V158()
V159()
V160() )
sequence of ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) ) ) ;
theorem
for
a,
b being ( (
real ) (
V11()
real ext-real )
number )
for
s being ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) st
rng s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) : ( ( ) (
V168()
V169()
V170() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
c= [.a : ( ( real ) ( V11() real ext-real ) number ) ,b : ( ( real ) ( V11() real ext-real ) number ) .] : ( ( ) (
V168()
V169()
V170() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) holds
s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) is ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) ;
theorem
for
a,
b being ( (
real ) (
V11()
real ext-real )
number )
for
S being ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) st
a : ( (
real ) (
V11()
real ext-real )
number )
<= b : ( (
real ) (
V11()
real ext-real )
number ) holds
S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) is ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) )
V158()
V159()
V160() )
sequence of ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) ;
theorem
for
a,
b being ( (
real ) (
V11()
real ext-real )
number )
for
S1 being ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) )
for
S being ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) )
V158()
V159()
V160() )
sequence of ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) st
S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) )
V158()
V159()
V160() )
sequence of ( ( ) ( non
empty V168()
V169()
V170() )
set ) )
= S1 : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) &
a : ( (
real ) (
V11()
real ext-real )
number )
<= b : ( (
real ) (
V11()
real ext-real )
number ) holds
( (
S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) )
V158()
V159()
V160() )
sequence of ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) is
convergent implies
S1 : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) is
convergent ) & (
S1 : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) is
convergent implies
S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) )
V158()
V159()
V160() )
sequence of ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) is
convergent ) & (
S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) )
V158()
V159()
V160() )
sequence of ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) is
convergent implies
lim S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) )
V158()
V159()
V160() )
sequence of ( ( ) ( non
empty V168()
V169()
V170() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of the
carrier of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) : ( ( ) ( non
empty V168()
V169()
V170() )
set ) )
= lim S1 : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) ) ;
theorem
for
a,
b being ( (
real ) (
V11()
real ext-real )
number )
for
s being ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence)
for
S being ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) st
S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) )
= s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) &
a : ( (
real ) (
V11()
real ext-real )
number )
<= b : ( (
real ) (
V11()
real ext-real )
number ) &
s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) is
convergent holds
(
S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) is
convergent &
lim s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
= lim S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) ;
theorem
for
a,
b being ( (
real ) (
V11()
real ext-real )
number )
for
s being ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence)
for
S being ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) st
S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) )
= s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) &
a : ( (
real ) (
V11()
real ext-real )
number )
<= b : ( (
real ) (
V11()
real ext-real )
number ) &
s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) is
non-decreasing holds
S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) is
convergent ;
theorem
for
a,
b being ( (
real ) (
V11()
real ext-real )
number )
for
s being ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence)
for
S being ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) st
S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) )
= s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) &
a : ( (
real ) (
V11()
real ext-real )
number )
<= b : ( (
real ) (
V11()
real ext-real )
number ) &
s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) is
non-increasing holds
S : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
(Closed-Interval-MSpace (b1 : ( ( real ) ( V11() real ext-real ) number ) ,b2 : ( ( real ) ( V11() real ext-real ) number ) )) : ( ( non
empty strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning )
SubSpace of
RealSpace : ( (
strict ) ( non
empty strict Reflexive discerning V86()
triangle Discerning V116() )
MetrStruct ) ) : ( ( ) ( non
empty )
set ) ) )
sequence of ( ( ) ( non
empty )
set ) ) is
convergent ;
theorem
for
R being ( ( non
empty ) ( non
empty V168()
V169()
V170() )
Subset of ( ( ) ( )
set ) ) st
R : ( ( non
empty ) ( non
empty V168()
V169()
V170() )
Subset of ( ( ) ( )
set ) ) is
bounded_above holds
ex
s being ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) st
(
s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) is
non-decreasing &
s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) is
convergent &
rng s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) : ( ( ) (
V168()
V169()
V170() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
c= R : ( ( non
empty ) ( non
empty V168()
V169()
V170() )
Subset of ( ( ) ( )
set ) ) &
lim s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
= upper_bound R : ( ( non
empty ) ( non
empty V168()
V169()
V170() )
Subset of ( ( ) ( )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) ;
theorem
for
R being ( ( non
empty ) ( non
empty V168()
V169()
V170() )
Subset of ( ( ) ( )
set ) ) st
R : ( ( non
empty ) ( non
empty V168()
V169()
V170() )
Subset of ( ( ) ( )
set ) ) is
bounded_below holds
ex
s being ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) st
(
s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) is
non-increasing &
s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) is
convergent &
rng s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) : ( ( ) (
V168()
V169()
V170() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
c= R : ( ( non
empty ) ( non
empty V168()
V169()
V170() )
Subset of ( ( ) ( )
set ) ) &
lim s : ( (
Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set )
-valued Function-like V33(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
V158()
V159()
V160() )
Real_Sequence) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) )
= lower_bound R : ( ( non
empty ) ( non
empty V168()
V169()
V170() )
Subset of ( ( ) ( )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) ) ;
theorem
for
X being ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace)
for
f being ( (
Function-like V33( the
carrier of
I[01] : ( ( ) (
TopSpace-like V116() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V116() )
TopStruct ) ) : ( ( ) (
V168()
V169()
V170() )
set ) , the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
I[01] : ( ( ) (
TopSpace-like V116() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V116() )
TopStruct ) ) : ( ( ) (
V168()
V169()
V170() )
set )
-defined the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set )
-valued Function-like V33( the
carrier of
I[01] : ( ( ) (
TopSpace-like V116() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V116() )
TopStruct ) ) : ( ( ) (
V168()
V169()
V170() )
set ) , the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) )
Function of ( ( ) (
V168()
V169()
V170() )
set ) , ( ( ) ( non
empty )
set ) )
for
F1,
F2 being ( ( ) ( )
Subset of )
for
r1,
r2 being ( ( ) (
V11()
real ext-real )
Real) st
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11()
real ext-real non
positive non
negative V120()
V121()
V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) )
<= r1 : ( ( ) (
V11()
real ext-real )
Real) &
r2 : ( ( ) (
V11()
real ext-real )
Real)
<= 1 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V120()
V121()
V168()
V169()
V170()
V171()
V172()
V173() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V168()
V169()
V170()
V171()
V172()
V173()
V174() )
Element of
K6(
REAL : ( ( ) ( non
empty V36()
V168()
V169()
V170()
V174() )
set ) ) : ( ( ) ( )
set ) ) ) &
r1 : ( ( ) (
V11()
real ext-real )
Real)
<= r2 : ( ( ) (
V11()
real ext-real )
Real) &
f : ( (
Function-like V33( the
carrier of
I[01] : ( ( ) (
TopSpace-like V116() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V116() )
TopStruct ) ) : ( ( ) (
V168()
V169()
V170() )
set ) , the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
I[01] : ( ( ) (
TopSpace-like V116() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V116() )
TopStruct ) ) : ( ( ) (
V168()
V169()
V170() )
set )
-defined the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set )
-valued Function-like V33( the
carrier of
I[01] : ( ( ) (
TopSpace-like V116() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V116() )
TopStruct ) ) : ( ( ) (
V168()
V169()
V170() )
set ) , the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) )
Function of ( ( ) (
V168()
V169()
V170() )
set ) , ( ( ) ( non
empty )
set ) )
. r1 : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) ( )
set )
in F1 : ( ( ) ( )
Subset of ) &
f : ( (
Function-like V33( the
carrier of
I[01] : ( ( ) (
TopSpace-like V116() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V116() )
TopStruct ) ) : ( ( ) (
V168()
V169()
V170() )
set ) , the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
I[01] : ( ( ) (
TopSpace-like V116() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V116() )
TopStruct ) ) : ( ( ) (
V168()
V169()
V170() )
set )
-defined the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set )
-valued Function-like V33( the
carrier of
I[01] : ( ( ) (
TopSpace-like V116() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V116() )
TopStruct ) ) : ( ( ) (
V168()
V169()
V170() )
set ) , the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) )
Function of ( ( ) (
V168()
V169()
V170() )
set ) , ( ( ) ( non
empty )
set ) )
. r2 : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) ( )
set )
in F2 : ( ( ) ( )
Subset of ) &
F1 : ( ( ) ( )
Subset of ) is
closed &
F2 : ( ( ) ( )
Subset of ) is
closed &
f : ( (
Function-like V33( the
carrier of
I[01] : ( ( ) (
TopSpace-like V116() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V116() )
TopStruct ) ) : ( ( ) (
V168()
V169()
V170() )
set ) , the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
I[01] : ( ( ) (
TopSpace-like V116() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V116() )
TopStruct ) ) : ( ( ) (
V168()
V169()
V170() )
set )
-defined the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set )
-valued Function-like V33( the
carrier of
I[01] : ( ( ) (
TopSpace-like V116() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V116() )
TopStruct ) ) : ( ( ) (
V168()
V169()
V170() )
set ) , the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) )
Function of ( ( ) (
V168()
V169()
V170() )
set ) , ( ( ) ( non
empty )
set ) ) is
continuous &
F1 : ( ( ) ( )
Subset of )
\/ F2 : ( ( ) ( )
Subset of ) : ( ( ) ( )
Element of
K6( the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
= the
carrier of
X : ( ( non
empty Reflexive discerning V86()
triangle ) ( non
empty Reflexive discerning V86()
triangle Discerning )
MetrSpace) : ( ( ) ( non
empty )
set ) holds
ex
r being ( ( ) (
V11()
real ext-real )
Real) st
(
r1 : ( ( ) (
V11()
real ext-real )
Real)
<= r : ( ( ) (
V11()
real ext-real )
Real) &
r : ( ( ) (
V11()
real ext-real )
Real)
<= r2 : ( ( ) (
V11()
real ext-real )
Real) &
f : ( (
Function-like V33( the
carrier of
I[01] : ( ( ) (
TopSpace-like V116() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V116() )
TopStruct ) ) : ( ( ) (
V168()
V169()
V170() )
set ) , the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
I[01] : ( ( ) (
TopSpace-like V116() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V116() )
TopStruct ) ) : ( ( ) (
V168()
V169()
V170() )
set )
-defined the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set )
-valued Function-like V33( the
carrier of
I[01] : ( ( ) (
TopSpace-like V116() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V116() )
TopStruct ) ) : ( ( ) (
V168()
V169()
V170() )
set ) , the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) )
Function of ( ( ) (
V168()
V169()
V170() )
set ) , ( ( ) ( non
empty )
set ) )
. r : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) ( )
set )
in F1 : ( ( ) ( )
Subset of )
/\ F2 : ( ( ) ( )
Subset of ) : ( ( ) ( )
Element of
K6( the
carrier of
(TopSpaceMetr b1 : ( ( non empty Reflexive discerning V86() triangle ) ( non empty Reflexive discerning V86() triangle Discerning ) MetrSpace) ) : ( ( ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) ) ;