begin
theorem
for
n being ( ( ) (
natural V11()
real ext-real V43()
V44()
V45()
V46()
V47()
V48()
V49()
V50()
bounded_below )
Element of
NAT : ( ( ) (
V45()
V46()
V47()
V48()
V49()
V50()
V51()
bounded_below )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) )
for
p1,
p2 being ( ( ) (
Relation-like Function-like V33()
V34()
V35()
V69(
b1 : ( ( ) (
natural V11()
real ext-real V43()
V44()
V45()
V46()
V47()
V48()
V49()
V50()
bounded_below )
Element of
NAT : ( ( ) (
V45()
V46()
V47()
V48()
V49()
V50()
V51()
bounded_below )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) ) )
V70() )
Point of ( ( ) ( non
empty functional )
set ) ) st
p1 : ( ( ) (
Relation-like Function-like V33()
V34()
V35()
V69(
b1 : ( ( ) (
natural V11()
real ext-real V43()
V44()
V45()
V46()
V47()
V48()
V49()
V50()
bounded_below )
Element of
NAT : ( ( ) (
V45()
V46()
V47()
V48()
V49()
V50()
V51()
bounded_below )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) ) )
V70() )
Point of ( ( ) ( non
empty functional )
set ) )
<> p2 : ( ( ) (
Relation-like Function-like V33()
V34()
V35()
V69(
b1 : ( ( ) (
natural V11()
real ext-real V43()
V44()
V45()
V46()
V47()
V48()
V49()
V50()
bounded_below )
Element of
NAT : ( ( ) (
V45()
V46()
V47()
V48()
V49()
V50()
V51()
bounded_below )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) ) )
V70() )
Point of ( ( ) ( non
empty functional )
set ) ) holds
ex
f being ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like T_0 T_1 T_2 compact V212()
pathwise_connected pseudocompact )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
((TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V146() V171() V172() V173() V174() V175() V176() V177() strict ) RLTopStruct ) | (LSeg (b2 : ( ( ) ( Relation-like Function-like V33() V34() V35() V69(b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V70() ) Point of ( ( ) ( non empty functional ) set ) ) ,b3 : ( ( ) ( Relation-like Function-like V33() V34() V35() V69(b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V70() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( non empty functional V231( TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V146() V171() V172() V173() V174() V175() V176() V177() strict ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V146() V171() V172() V173() V174() V175() V176() V177() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( (
strict ) ( non
empty strict TopSpace-like T_0 T_1 T_2 )
SubSpace of
TOP-REAL b1 : ( ( ) (
natural V11()
real ext-real V43()
V44()
V45()
V46()
V47()
V48()
V49()
V50()
bounded_below )
Element of
NAT : ( ( ) (
V45()
V46()
V47()
V48()
V49()
V50()
V51()
bounded_below )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) ) : ( (
strict ) ( non
empty TopSpace-like T_0 T_1 T_2 V146()
V171()
V172()
V173()
V174()
V175()
V176()
V177()
strict )
RLTopStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like total quasi_total )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty )
set ) ) st
( ( for
x being ( ( ) (
V11()
real ext-real )
Real) st
x : ( ( ) (
V11()
real ext-real )
Real)
in [.0 : ( ( ) ( empty natural V11() real ext-real non positive non negative Function-like functional V43() V44() V45() V46() V47() V48() V49() V50() V51() finite V56() bounded_below bounded_above real-bounded V66() ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V49() V50() left_end bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) (
V45()
V46()
V47()
V66()
closed )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) holds
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like T_0 T_1 T_2 compact V212()
pathwise_connected pseudocompact )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
((TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V146() V171() V172() V173() V174() V175() V176() V177() strict ) RLTopStruct ) | (LSeg (b2 : ( ( ) ( Relation-like Function-like V33() V34() V35() V69(b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V70() ) Point of ( ( ) ( non empty functional ) set ) ) ,b3 : ( ( ) ( Relation-like Function-like V33() V34() V35() V69(b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V70() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( non empty functional V231( TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V146() V171() V172() V173() V174() V175() V176() V177() strict ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V146() V171() V172() V173() V174() V175() V176() V177() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( (
strict ) ( non
empty strict TopSpace-like T_0 T_1 T_2 )
SubSpace of
TOP-REAL b1 : ( ( ) (
natural V11()
real ext-real V43()
V44()
V45()
V46()
V47()
V48()
V49()
V50()
bounded_below )
Element of
NAT : ( ( ) (
V45()
V46()
V47()
V48()
V49()
V50()
V51()
bounded_below )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) ) : ( (
strict ) ( non
empty TopSpace-like T_0 T_1 T_2 V146()
V171()
V172()
V173()
V174()
V175()
V176()
V177()
strict )
RLTopStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like total quasi_total )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty )
set ) )
. x : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) ( )
set )
= ((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V49() V50() left_end bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) - x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) * p1 : ( ( ) ( Relation-like Function-like V33() V34() V35() V69(b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V70() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) (
Relation-like Function-like V33()
V34()
V35()
V69(
b1 : ( ( ) (
natural V11()
real ext-real V43()
V44()
V45()
V46()
V47()
V48()
V49()
V50()
bounded_below )
Element of
NAT : ( ( ) (
V45()
V46()
V47()
V48()
V49()
V50()
V51()
bounded_below )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) ) )
V70() )
Element of the
carrier of
(TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like T_0 T_1 T_2 V146()
V171()
V172()
V173()
V174()
V175()
V176()
V177()
strict )
RLTopStruct ) : ( ( ) ( non
empty functional )
set ) )
+ (x : ( ( ) ( V11() real ext-real ) Real) * p2 : ( ( ) ( Relation-like Function-like V33() V34() V35() V69(b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V70() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) (
Relation-like Function-like V33()
V34()
V35()
V69(
b1 : ( ( ) (
natural V11()
real ext-real V43()
V44()
V45()
V46()
V47()
V48()
V49()
V50()
bounded_below )
Element of
NAT : ( ( ) (
V45()
V46()
V47()
V48()
V49()
V50()
V51()
bounded_below )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) ) )
V70() )
Element of the
carrier of
(TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like T_0 T_1 T_2 V146()
V171()
V172()
V173()
V174()
V175()
V176()
V177()
strict )
RLTopStruct ) : ( ( ) ( non
empty functional )
set ) ) : ( ( ) (
Relation-like Function-like V33()
V34()
V35()
V69(
b1 : ( ( ) (
natural V11()
real ext-real V43()
V44()
V45()
V46()
V47()
V48()
V49()
V50()
bounded_below )
Element of
NAT : ( ( ) (
V45()
V46()
V47()
V48()
V49()
V50()
V51()
bounded_below )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) ) )
V70() )
Element of the
carrier of
(TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like T_0 T_1 T_2 V146()
V171()
V172()
V173()
V174()
V175()
V176()
V177()
strict )
RLTopStruct ) : ( ( ) ( non
empty functional )
set ) ) ) &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like T_0 T_1 T_2 compact V212()
pathwise_connected pseudocompact )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
((TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V146() V171() V172() V173() V174() V175() V176() V177() strict ) RLTopStruct ) | (LSeg (b2 : ( ( ) ( Relation-like Function-like V33() V34() V35() V69(b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V70() ) Point of ( ( ) ( non empty functional ) set ) ) ,b3 : ( ( ) ( Relation-like Function-like V33() V34() V35() V69(b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V70() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( non empty functional V231( TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V146() V171() V172() V173() V174() V175() V176() V177() strict ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V146() V171() V172() V173() V174() V175() V176() V177() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( (
strict ) ( non
empty strict TopSpace-like T_0 T_1 T_2 )
SubSpace of
TOP-REAL b1 : ( ( ) (
natural V11()
real ext-real V43()
V44()
V45()
V46()
V47()
V48()
V49()
V50()
bounded_below )
Element of
NAT : ( ( ) (
V45()
V46()
V47()
V48()
V49()
V50()
V51()
bounded_below )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) ) : ( (
strict ) ( non
empty TopSpace-like T_0 T_1 T_2 V146()
V171()
V172()
V173()
V174()
V175()
V176()
V177()
strict )
RLTopStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like total quasi_total )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty )
set ) ) is
being_homeomorphism &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like T_0 T_1 T_2 compact V212()
pathwise_connected pseudocompact )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
((TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V146() V171() V172() V173() V174() V175() V176() V177() strict ) RLTopStruct ) | (LSeg (b2 : ( ( ) ( Relation-like Function-like V33() V34() V35() V69(b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V70() ) Point of ( ( ) ( non empty functional ) set ) ) ,b3 : ( ( ) ( Relation-like Function-like V33() V34() V35() V69(b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V70() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( non empty functional V231( TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V146() V171() V172() V173() V174() V175() V176() V177() strict ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V146() V171() V172() V173() V174() V175() V176() V177() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( (
strict ) ( non
empty strict TopSpace-like T_0 T_1 T_2 )
SubSpace of
TOP-REAL b1 : ( ( ) (
natural V11()
real ext-real V43()
V44()
V45()
V46()
V47()
V48()
V49()
V50()
bounded_below )
Element of
NAT : ( ( ) (
V45()
V46()
V47()
V48()
V49()
V50()
V51()
bounded_below )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) ) : ( (
strict ) ( non
empty TopSpace-like T_0 T_1 T_2 V146()
V171()
V172()
V173()
V174()
V175()
V176()
V177()
strict )
RLTopStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like total quasi_total )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty )
set ) )
. 0 : ( ( ) (
empty natural V11()
real ext-real non
positive non
negative Function-like functional V43()
V44()
V45()
V46()
V47()
V48()
V49()
V50()
V51()
finite V56()
bounded_below bounded_above real-bounded V66() )
Element of
NAT : ( ( ) (
V45()
V46()
V47()
V48()
V49()
V50()
V51()
bounded_below )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
set )
= p1 : ( ( ) (
Relation-like Function-like V33()
V34()
V35()
V69(
b1 : ( ( ) (
natural V11()
real ext-real V43()
V44()
V45()
V46()
V47()
V48()
V49()
V50()
bounded_below )
Element of
NAT : ( ( ) (
V45()
V46()
V47()
V48()
V49()
V50()
V51()
bounded_below )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) ) )
V70() )
Point of ( ( ) ( non
empty functional )
set ) ) &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like T_0 T_1 T_2 compact V212()
pathwise_connected pseudocompact )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
((TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V146() V171() V172() V173() V174() V175() V176() V177() strict ) RLTopStruct ) | (LSeg (b2 : ( ( ) ( Relation-like Function-like V33() V34() V35() V69(b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V70() ) Point of ( ( ) ( non empty functional ) set ) ) ,b3 : ( ( ) ( Relation-like Function-like V33() V34() V35() V69(b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V70() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( non empty functional V231( TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V146() V171() V172() V173() V174() V175() V176() V177() strict ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( natural V11() real ext-real V43() V44() V45() V46() V47() V48() V49() V50() bounded_below ) Element of NAT : ( ( ) ( V45() V46() V47() V48() V49() V50() V51() bounded_below ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V146() V171() V172() V173() V174() V175() V176() V177() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( (
strict ) ( non
empty strict TopSpace-like T_0 T_1 T_2 )
SubSpace of
TOP-REAL b1 : ( ( ) (
natural V11()
real ext-real V43()
V44()
V45()
V46()
V47()
V48()
V49()
V50()
bounded_below )
Element of
NAT : ( ( ) (
V45()
V46()
V47()
V48()
V49()
V50()
V51()
bounded_below )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) ) : ( (
strict ) ( non
empty TopSpace-like T_0 T_1 T_2 V146()
V171()
V172()
V173()
V174()
V175()
V176()
V177()
strict )
RLTopStruct ) ) : ( ( ) ( non
empty )
set )
-valued Function-like total quasi_total )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty )
set ) )
. 1 : ( ( ) ( non
empty natural V11()
real ext-real positive non
negative V43()
V44()
V45()
V46()
V47()
V48()
V49()
V50()
left_end bounded_below )
Element of
NAT : ( ( ) (
V45()
V46()
V47()
V48()
V49()
V50()
V51()
bounded_below )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
set )
= p2 : ( ( ) (
Relation-like Function-like V33()
V34()
V35()
V69(
b1 : ( ( ) (
natural V11()
real ext-real V43()
V44()
V45()
V46()
V47()
V48()
V49()
V50()
bounded_below )
Element of
NAT : ( ( ) (
V45()
V46()
V47()
V48()
V49()
V50()
V51()
bounded_below )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) ) )
V70() )
Point of ( ( ) ( non
empty functional )
set ) ) ) ;
begin
theorem
for
f being ( (
Function-like one-to-one quasi_total continuous ) ( non
empty Relation-like the
carrier of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like one-to-one total quasi_total V33()
V34()
V35()
continuous )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) holds
( for
x,
y being ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
for
p,
q,
fx,
fy being ( ( ) (
V11()
real ext-real )
Real) st
x : ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
= p : ( ( ) (
V11()
real ext-real )
Real) &
y : ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
= q : ( ( ) (
V11()
real ext-real )
Real) &
p : ( ( ) (
V11()
real ext-real )
Real)
< q : ( ( ) (
V11()
real ext-real )
Real) &
fx : ( ( ) (
V11()
real ext-real )
Real)
= f : ( (
Function-like one-to-one quasi_total continuous ) ( non
empty Relation-like the
carrier of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like one-to-one total quasi_total V33()
V34()
V35()
continuous )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. x : ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) : ( ( ) (
V11()
real ext-real )
set ) &
fy : ( ( ) (
V11()
real ext-real )
Real)
= f : ( (
Function-like one-to-one quasi_total continuous ) ( non
empty Relation-like the
carrier of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like one-to-one total quasi_total V33()
V34()
V35()
continuous )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. y : ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) : ( ( ) (
V11()
real ext-real )
set ) holds
fx : ( ( ) (
V11()
real ext-real )
Real)
< fy : ( ( ) (
V11()
real ext-real )
Real) or for
x,
y being ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
for
p,
q,
fx,
fy being ( ( ) (
V11()
real ext-real )
Real) st
x : ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
= p : ( ( ) (
V11()
real ext-real )
Real) &
y : ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
= q : ( ( ) (
V11()
real ext-real )
Real) &
p : ( ( ) (
V11()
real ext-real )
Real)
< q : ( ( ) (
V11()
real ext-real )
Real) &
fx : ( ( ) (
V11()
real ext-real )
Real)
= f : ( (
Function-like one-to-one quasi_total continuous ) ( non
empty Relation-like the
carrier of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like one-to-one total quasi_total V33()
V34()
V35()
continuous )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. x : ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) : ( ( ) (
V11()
real ext-real )
set ) &
fy : ( ( ) (
V11()
real ext-real )
Real)
= f : ( (
Function-like one-to-one quasi_total continuous ) ( non
empty Relation-like the
carrier of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like one-to-one total quasi_total V33()
V34()
V35()
continuous )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. y : ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) : ( ( ) (
V11()
real ext-real )
set ) holds
fx : ( ( ) (
V11()
real ext-real )
Real)
> fy : ( ( ) (
V11()
real ext-real )
Real) ) ;
theorem
for
a,
b,
c,
d,
x1 being ( ( ) (
V11()
real ext-real )
Real)
for
f being ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
for
x being ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
for
g being ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set )
-defined REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set )
-valued Function-like V33()
V34()
V35() )
PartFunc of ,) st
a : ( ( ) (
V11()
real ext-real )
Real)
< b : ( ( ) (
V11()
real ext-real )
Real) &
c : ( ( ) (
V11()
real ext-real )
Real)
< d : ( ( ) (
V11()
real ext-real )
Real) &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
is_continuous_at x : ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. a : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
set )
= c : ( ( ) (
V11()
real ext-real )
Real) &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. b : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
set )
= d : ( ( ) (
V11()
real ext-real )
Real) &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) is
one-to-one &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
= g : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set )
-defined REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set )
-valued Function-like V33()
V34()
V35() )
PartFunc of ,) &
x : ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
= x1 : ( ( ) (
V11()
real ext-real )
Real) holds
g : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set )
-defined REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set )
-valued Function-like V33()
V34()
V35() )
PartFunc of ,)
| [.a : ( ( ) ( V11() real ext-real ) Real) ,b : ( ( ) ( V11() real ext-real ) Real) .] : ( ( ) (
V45()
V46()
V47()
V66()
closed )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set )
-defined REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set )
-valued Function-like V33()
V34()
V35() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ,
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty V33()
V34()
V35() )
set ) ) : ( ( ) ( non
empty )
set ) )
is_continuous_in x1 : ( ( ) (
V11()
real ext-real )
Real) ;
theorem
for
a,
b,
c,
d being ( ( ) (
V11()
real ext-real )
Real)
for
f being ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
for
g being ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set )
-defined REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set )
-valued Function-like V33()
V34()
V35() )
PartFunc of ,) st
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) is
continuous &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) is
one-to-one &
a : ( ( ) (
V11()
real ext-real )
Real)
< b : ( ( ) (
V11()
real ext-real )
Real) &
c : ( ( ) (
V11()
real ext-real )
Real)
< d : ( ( ) (
V11()
real ext-real )
Real) &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
= g : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set )
-defined REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set )
-valued Function-like V33()
V34()
V35() )
PartFunc of ,) &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. a : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
set )
= c : ( ( ) (
V11()
real ext-real )
Real) &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. b : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
set )
= d : ( ( ) (
V11()
real ext-real )
Real) holds
g : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set )
-defined REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set )
-valued Function-like V33()
V34()
V35() )
PartFunc of ,)
| [.a : ( ( ) ( V11() real ext-real ) Real) ,b : ( ( ) ( V11() real ext-real ) Real) .] : ( ( ) (
V45()
V46()
V47()
V66()
closed )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set )
-defined REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set )
-valued Function-like V33()
V34()
V35() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ,
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty V33()
V34()
V35() )
set ) ) : ( ( ) ( non
empty )
set ) ) is
continuous ;
begin
theorem
for
a,
b,
c,
d being ( ( ) (
V11()
real ext-real )
Real)
for
f being ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) st
a : ( ( ) (
V11()
real ext-real )
Real)
< b : ( ( ) (
V11()
real ext-real )
Real) &
c : ( ( ) (
V11()
real ext-real )
Real)
< d : ( ( ) (
V11()
real ext-real )
Real) &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) is
continuous &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) is
one-to-one &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. a : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
set )
= c : ( ( ) (
V11()
real ext-real )
Real) &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. b : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
set )
= d : ( ( ) (
V11()
real ext-real )
Real) holds
for
x,
y being ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
for
p,
q,
fx,
fy being ( ( ) (
V11()
real ext-real )
Real) st
x : ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
= p : ( ( ) (
V11()
real ext-real )
Real) &
y : ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
= q : ( ( ) (
V11()
real ext-real )
Real) &
p : ( ( ) (
V11()
real ext-real )
Real)
< q : ( ( ) (
V11()
real ext-real )
Real) &
fx : ( ( ) (
V11()
real ext-real )
Real)
= f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. x : ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) &
fy : ( ( ) (
V11()
real ext-real )
Real)
= f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. y : ( ( ) (
V11()
real ext-real )
Point of ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) holds
fx : ( ( ) (
V11()
real ext-real )
Real)
< fy : ( ( ) (
V11()
real ext-real )
Real) ;
theorem
for
a,
b,
c,
d being ( ( ) (
V11()
real ext-real )
Real)
for
f being ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
for
P being ( ( non
empty ) ( non
empty V45()
V46()
V47() )
Subset of )
for
PP,
QQ being ( ( ) (
V45()
V46()
V47() )
Subset of ) st
a : ( ( ) (
V11()
real ext-real )
Real)
< b : ( ( ) (
V11()
real ext-real )
Real) &
c : ( ( ) (
V11()
real ext-real )
Real)
< d : ( ( ) (
V11()
real ext-real )
Real) &
PP : ( ( ) (
V45()
V46()
V47() )
Subset of )
= P : ( ( non
empty ) ( non
empty V45()
V46()
V47() )
Subset of ) &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) is
continuous &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) is
one-to-one &
PP : ( ( ) (
V45()
V46()
V47() )
Subset of ) is
compact &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. a : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
set )
= c : ( ( ) (
V11()
real ext-real )
Real) &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. b : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
set )
= d : ( ( ) (
V11()
real ext-real )
Real) &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
.: P : ( ( non
empty ) ( non
empty V45()
V46()
V47() )
Subset of ) : ( ( ) (
V45()
V46()
V47() )
Element of
K6( the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) : ( ( ) ( non
empty )
set ) )
= QQ : ( ( ) (
V45()
V46()
V47() )
Subset of ) holds
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. (lower_bound ([#] PP : ( ( ) ( V45() V46() V47() ) Subset of ) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) (
V11()
real ext-real )
set )
= lower_bound ([#] QQ : ( ( ) ( V45() V46() V47() ) Subset of ) ) : ( ( ) (
V45()
V46()
V47() )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) ;
theorem
for
a,
b,
c,
d being ( ( ) (
V11()
real ext-real )
Real)
for
f being ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
for
P,
Q being ( ( non
empty ) ( non
empty V45()
V46()
V47() )
Subset of )
for
PP,
QQ being ( ( ) (
V45()
V46()
V47() )
Subset of ) st
a : ( ( ) (
V11()
real ext-real )
Real)
< b : ( ( ) (
V11()
real ext-real )
Real) &
c : ( ( ) (
V11()
real ext-real )
Real)
< d : ( ( ) (
V11()
real ext-real )
Real) &
PP : ( ( ) (
V45()
V46()
V47() )
Subset of )
= P : ( ( non
empty ) ( non
empty V45()
V46()
V47() )
Subset of ) &
QQ : ( ( ) (
V45()
V46()
V47() )
Subset of )
= Q : ( ( non
empty ) ( non
empty V45()
V46()
V47() )
Subset of ) &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) is
continuous &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) is
one-to-one &
PP : ( ( ) (
V45()
V46()
V47() )
Subset of ) is
compact &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. a : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
set )
= c : ( ( ) (
V11()
real ext-real )
Real) &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. b : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
set )
= d : ( ( ) (
V11()
real ext-real )
Real) &
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
.: P : ( ( non
empty ) ( non
empty V45()
V46()
V47() )
Subset of ) : ( ( ) (
V45()
V46()
V47() )
Element of
K6( the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) : ( ( ) ( non
empty )
set ) )
= Q : ( ( non
empty ) ( non
empty V45()
V46()
V47() )
Subset of ) holds
f : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. (upper_bound ([#] PP : ( ( ) ( V45() V46() V47() ) Subset of ) ) : ( ( ) ( V45() V46() V47() ) Element of K6(REAL : ( ( ) ( non empty V2() V45() V46() V47() V51() non finite non bounded_below non bounded_above V66() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) (
V11()
real ext-real )
set )
= upper_bound ([#] QQ : ( ( ) ( V45() V46() V47() ) Subset of ) ) : ( ( ) (
V45()
V46()
V47() )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) ;
theorem
for
a,
b,
c,
d,
e,
f,
g,
h being ( ( ) (
V11()
real ext-real )
Real)
for
F being ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) st
a : ( ( ) (
V11()
real ext-real )
Real)
< b : ( ( ) (
V11()
real ext-real )
Real) &
c : ( ( ) (
V11()
real ext-real )
Real)
< d : ( ( ) (
V11()
real ext-real )
Real) &
e : ( ( ) (
V11()
real ext-real )
Real)
< f : ( ( ) (
V11()
real ext-real )
Real) &
a : ( ( ) (
V11()
real ext-real )
Real)
<= e : ( ( ) (
V11()
real ext-real )
Real) &
f : ( ( ) (
V11()
real ext-real )
Real)
<= b : ( ( ) (
V11()
real ext-real )
Real) &
F : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) is
being_homeomorphism &
F : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. a : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
set )
= c : ( ( ) (
V11()
real ext-real )
Real) &
F : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. b : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
set )
= d : ( ( ) (
V11()
real ext-real )
Real) &
g : ( ( ) (
V11()
real ext-real )
Real)
= F : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. e : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
set ) &
h : ( ( ) (
V11()
real ext-real )
Real)
= F : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
. f : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
set ) holds
F : ( (
Function-like quasi_total ) ( non
empty Relation-like the
carrier of
(Closed-Interval-TSpace (b1 : ( ( ) ( V11() real ext-real ) Real) ,b2 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-defined the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set )
-valued Function-like total quasi_total V33()
V34()
V35() )
Function of ( ( ) ( non
empty V45()
V46()
V47() )
set ) , ( ( ) ( non
empty V45()
V46()
V47() )
set ) )
.: [.e : ( ( ) ( V11() real ext-real ) Real) ,f : ( ( ) ( V11() real ext-real ) Real) .] : ( ( ) (
V45()
V46()
V47()
V66()
closed )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) : ( ( ) (
V45()
V46()
V47() )
Element of
K6( the
carrier of
(Closed-Interval-TSpace (b3 : ( ( ) ( V11() real ext-real ) Real) ,b4 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( non
empty strict ) ( non
empty strict TopSpace-like V212() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V212() )
TopStruct ) ) : ( ( ) ( non
empty V45()
V46()
V47() )
set ) ) : ( ( ) ( non
empty )
set ) )
= [.g : ( ( ) ( V11() real ext-real ) Real) ,h : ( ( ) ( V11() real ext-real ) Real) .] : ( ( ) (
V45()
V46()
V47()
V66()
closed )
Element of
K6(
REAL : ( ( ) ( non
empty V2()
V45()
V46()
V47()
V51() non
finite non
bounded_below non
bounded_above V66() )
set ) ) : ( ( ) ( non
empty )
set ) ) ;