begin
theorem
for
p,
g being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,) st (
f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,)
| ].p : ( ( ) ( V22() real ext-real ) Real) ,g : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) (
open V84()
V85()
V86()
V95()
V96()
V100() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
].b1 : ( ( ) ( V22() real ext-real ) Real) ,b2 : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) (
open V84()
V85()
V86()
V95()
V96()
V100() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
increasing or
f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,)
| ].p : ( ( ) ( V22() real ext-real ) Real) ,g : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) (
open V84()
V85()
V86()
V95()
V96()
V100() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
].b1 : ( ( ) ( V22() real ext-real ) Real) ,b2 : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) (
open V84()
V85()
V86()
V95()
V96()
V100() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
decreasing ) &
].p : ( ( ) ( V22() real ext-real ) Real) ,g : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) (
open V84()
V85()
V86()
V95()
V96()
V100() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) holds
((f : ( ( Function-like one-to-one ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like one-to-one V33() V34() V35() ) PartFunc of ,) | ].p : ( ( ) ( V22() real ext-real ) Real) ,g : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) ( open V84() V85() V86() V95() V96() V100() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V4(].b1 : ( ( ) ( V22() real ext-real ) Real) ,b2 : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) ( open V84() V85() V86() V95() V96() V100() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like one-to-one V33() V34() V35() ) Element of K19(K20(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ,REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( V1() V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ") : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
| (f : ( ( Function-like one-to-one ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like one-to-one V33() V34() V35() ) PartFunc of ,) .: ].p : ( ( ) ( V22() real ext-real ) Real) ,g : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) ( open V84() V85() V86() V95() V96() V100() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
b3 : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,)
.: ].b1 : ( ( ) ( V22() real ext-real ) Real) ,b2 : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) (
open V84()
V85()
V86()
V95()
V96()
V100() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
continuous ;
theorem
for
p being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,) st (
f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,)
| (left_open_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
left_open_halfline b1 : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
increasing or
f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,)
| (left_open_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
left_open_halfline b1 : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
decreasing ) &
left_open_halfline p : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) holds
((f : ( ( Function-like one-to-one ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like one-to-one V33() V34() V35() ) PartFunc of ,) | (left_open_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( open V84() V85() V86() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V4( left_open_halfline b1 : ( ( ) ( V22() real ext-real ) Real) : ( ( ) ( open V84() V85() V86() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like one-to-one V33() V34() V35() ) Element of K19(K20(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ,REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( V1() V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ") : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
| (f : ( ( Function-like one-to-one ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like one-to-one V33() V34() V35() ) PartFunc of ,) .: (left_open_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( open V84() V85() V86() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
b2 : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,)
.: (left_open_halfline b1 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
continuous ;
theorem
for
p being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,) st (
f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,)
| (right_open_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
right_open_halfline b1 : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
increasing or
f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,)
| (right_open_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
right_open_halfline b1 : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
decreasing ) &
right_open_halfline p : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) holds
((f : ( ( Function-like one-to-one ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like one-to-one V33() V34() V35() ) PartFunc of ,) | (right_open_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( open V84() V85() V86() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V4( right_open_halfline b1 : ( ( ) ( V22() real ext-real ) Real) : ( ( ) ( open V84() V85() V86() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like one-to-one V33() V34() V35() ) Element of K19(K20(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ,REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( V1() V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ") : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
| (f : ( ( Function-like one-to-one ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like one-to-one V33() V34() V35() ) PartFunc of ,) .: (right_open_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( open V84() V85() V86() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
b2 : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,)
.: (right_open_halfline b1 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
continuous ;
theorem
for
p being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,) st (
f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,)
| (left_closed_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
closed V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
left_closed_halfline b1 : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
closed V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
increasing or
f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,)
| (left_closed_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
closed V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
left_closed_halfline b1 : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
closed V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
decreasing ) &
left_closed_halfline p : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
closed V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) holds
((f : ( ( Function-like one-to-one ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like one-to-one V33() V34() V35() ) PartFunc of ,) | (left_closed_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( closed V84() V85() V86() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V4( left_closed_halfline b1 : ( ( ) ( V22() real ext-real ) Real) : ( ( ) ( closed V84() V85() V86() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like one-to-one V33() V34() V35() ) Element of K19(K20(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ,REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( V1() V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ") : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
| (f : ( ( Function-like one-to-one ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like one-to-one V33() V34() V35() ) PartFunc of ,) .: (left_closed_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( closed V84() V85() V86() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
b2 : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,)
.: (left_closed_halfline b1 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
closed V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
continuous ;
theorem
for
p being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,) st (
f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,)
| (right_closed_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
closed V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
right_closed_halfline b1 : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
closed V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
increasing or
f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,)
| (right_closed_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
closed V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
right_closed_halfline b1 : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
closed V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
decreasing ) &
right_closed_halfline p : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
closed V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) holds
((f : ( ( Function-like one-to-one ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like one-to-one V33() V34() V35() ) PartFunc of ,) | (right_closed_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( closed V84() V85() V86() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V4( right_closed_halfline b1 : ( ( ) ( V22() real ext-real ) Real) : ( ( ) ( closed V84() V85() V86() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like one-to-one V33() V34() V35() ) Element of K19(K20(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ,REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( V1() V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ") : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
| (f : ( ( Function-like one-to-one ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like one-to-one V33() V34() V35() ) PartFunc of ,) .: (right_closed_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( closed V84() V85() V86() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
b2 : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,)
.: (right_closed_halfline b1 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
closed V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
continuous ;
theorem
for
f being ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,) st (
f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,)
| ([#] REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) (
closed open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
[#] REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) : ( ( ) (
closed open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
increasing or
f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,)
| ([#] REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) (
closed open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
[#] REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) : ( ( ) (
closed open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
decreasing ) &
f : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,) is
total holds
(f : ( ( Function-like one-to-one ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like one-to-one V33() V34() V35() ) PartFunc of ,) ") : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
| (rng f : ( ( Function-like one-to-one ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like one-to-one V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
rng b1 : ( (
Function-like one-to-one ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like one-to-one V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
continuous ;
theorem
for
p,
g being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
PartFunc of ,) st
].p : ( ( ) ( V22() real ext-real ) Real) ,g : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) (
open V84()
V85()
V86()
V95()
V96()
V100() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
PartFunc of ,)
| ].p : ( ( ) ( V22() real ext-real ) Real) ,g : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) (
open V84()
V85()
V86()
V95()
V96()
V100() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
].b1 : ( ( ) ( V22() real ext-real ) Real) ,b2 : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) (
open V84()
V85()
V86()
V95()
V96()
V100() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
continuous & (
f : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
PartFunc of ,)
| ].p : ( ( ) ( V22() real ext-real ) Real) ,g : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) (
open V84()
V85()
V86()
V95()
V96()
V100() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
].b1 : ( ( ) ( V22() real ext-real ) Real) ,b2 : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) (
open V84()
V85()
V86()
V95()
V96()
V100() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
increasing or
f : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
PartFunc of ,)
| ].p : ( ( ) ( V22() real ext-real ) Real) ,g : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) (
open V84()
V85()
V86()
V95()
V96()
V100() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
].b1 : ( ( ) ( V22() real ext-real ) Real) ,b2 : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) (
open V84()
V85()
V86()
V95()
V96()
V100() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
decreasing ) holds
rng (f : ( ( Function-like ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like V33() V34() V35() ) PartFunc of ,) | ].p : ( ( ) ( V22() real ext-real ) Real) ,g : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) ( open V84() V85() V86() V95() V96() V100() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
].b1 : ( ( ) ( V22() real ext-real ) Real) ,b2 : ( ( ) ( V22() real ext-real ) Real) .[ : ( ( ) (
open V84()
V85()
V86()
V95()
V96()
V100() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) is
open ;
theorem
for
p being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
PartFunc of ,) st
left_open_halfline p : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
PartFunc of ,)
| (left_open_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
left_open_halfline b1 : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
continuous & (
f : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
PartFunc of ,)
| (left_open_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
left_open_halfline b1 : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
increasing or
f : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
PartFunc of ,)
| (left_open_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
left_open_halfline b1 : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
decreasing ) holds
rng (f : ( ( Function-like ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like V33() V34() V35() ) PartFunc of ,) | (left_open_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( open V84() V85() V86() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
left_open_halfline b1 : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) is
open ;
theorem
for
p being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
PartFunc of ,) st
right_open_halfline p : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
PartFunc of ,)
| (right_open_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
right_open_halfline b1 : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
continuous & (
f : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
PartFunc of ,)
| (right_open_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
right_open_halfline b1 : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
increasing or
f : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
PartFunc of ,)
| (right_open_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
right_open_halfline b1 : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
decreasing ) holds
rng (f : ( ( Function-like ) ( V1() V4( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) V5( REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) Function-like V33() V34() V35() ) PartFunc of ,) | (right_open_halfline p : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( open V84() V85() V86() ) Element of K19(REAL : ( ( ) ( non empty V48() V84() V85() V86() V90() V97() V98() V100() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like ) (
V1()
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V4(
right_open_halfline b1 : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
open V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) )
Function-like V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ,
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) (
V1()
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K19(
REAL : ( ( ) ( non
empty V48()
V84()
V85()
V86()
V90()
V97()
V98()
V100() )
set ) ) : ( ( ) ( )
set ) ) is
open ;