begin
theorem
for
A being ( ( ) ( )
Subset of )
for
q1,
q2,
p1,
p2 being ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) st
LE q1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
q2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
A : ( ( ) ( )
Subset of ) ,
p1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) holds
(
q1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) )
in Segment (
A : ( ( ) ( )
Subset of ) ,
p1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
q1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
q2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K6( the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V118() V119() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non empty V38() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V190() ) ( non
empty TopSpace-like V132()
V178()
V179()
V180()
V181()
V182()
V183()
V184()
V190() )
L15()) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
q2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) )
in Segment (
A : ( ( ) ( )
Subset of ) ,
p1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
q1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
q2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K6( the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V118() V119() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non empty V38() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V190() ) ( non
empty TopSpace-like V132()
V178()
V179()
V180()
V181()
V182()
V183()
V184()
V190() )
L15()) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
A being ( ( ) ( )
Subset of )
for
p1,
p2,
q1,
q2 being ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) st
A : ( ( ) ( )
Subset of )
is_an_arc_of p1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) &
LE q1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
q2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
A : ( ( ) ( )
Subset of ) ,
p1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) holds
ex
g being ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V111() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V111() )
TopStruct ) ) : ( ( ) ( non
empty V166()
V167()
V168() )
set )
-defined the
carrier of
((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V118() V119() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non empty V38() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V190() ) ( non empty TopSpace-like V132() V178() V179() V180() V181() V182() V183() V184() V190() ) L15()) | b1 : ( ( ) ( ) Subset of ) ) : ( (
strict ) (
strict TopSpace-like )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V190() ) ( non
empty TopSpace-like V132()
V178()
V179()
V180()
V181()
V182()
V183()
V184()
V190() )
L15()) ) : ( ( ) ( )
set )
-valued Function-like quasi_total )
Function of ( ( ) ( non
empty V166()
V167()
V168() )
set ) , ( ( ) ( )
set ) ) ex
s1,
s2 being ( ( ) (
V11()
V12()
ext-real )
Real) st
(
g : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V111() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V111() )
TopStruct ) ) : ( ( ) ( non
empty V166()
V167()
V168() )
set )
-defined the
carrier of
((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V118() V119() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non empty V38() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V190() ) ( non empty TopSpace-like V132() V178() V179() V180() V181() V182() V183() V184() V190() ) L15()) | b1 : ( ( ) ( ) Subset of ) ) : ( (
strict ) (
strict TopSpace-like )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V190() ) ( non
empty TopSpace-like V132()
V178()
V179()
V180()
V181()
V182()
V183()
V184()
V190() )
L15()) ) : ( ( ) ( )
set )
-valued Function-like quasi_total )
Function of ( ( ) ( non
empty V166()
V167()
V168() )
set ) , ( ( ) ( )
set ) ) is
being_homeomorphism &
g : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V111() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V111() )
TopStruct ) ) : ( ( ) ( non
empty V166()
V167()
V168() )
set )
-defined the
carrier of
((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V118() V119() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non empty V38() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V190() ) ( non empty TopSpace-like V132() V178() V179() V180() V181() V182() V183() V184() V190() ) L15()) | b1 : ( ( ) ( ) Subset of ) ) : ( (
strict ) (
strict TopSpace-like )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V190() ) ( non
empty TopSpace-like V132()
V178()
V179()
V180()
V181()
V182()
V183()
V184()
V190() )
L15()) ) : ( ( ) ( )
set )
-valued Function-like quasi_total )
Function of ( ( ) ( non
empty V166()
V167()
V168() )
set ) , ( ( ) ( )
set ) )
. 0 : ( ( ) (
empty V10()
V11()
V12()
ext-real non
positive non
negative Function-like functional V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set )
= p1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) &
g : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V111() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V111() )
TopStruct ) ) : ( ( ) ( non
empty V166()
V167()
V168() )
set )
-defined the
carrier of
((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V118() V119() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non empty V38() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V190() ) ( non empty TopSpace-like V132() V178() V179() V180() V181() V182() V183() V184() V190() ) L15()) | b1 : ( ( ) ( ) Subset of ) ) : ( (
strict ) (
strict TopSpace-like )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V190() ) ( non
empty TopSpace-like V132()
V178()
V179()
V180()
V181()
V182()
V183()
V184()
V190() )
L15()) ) : ( ( ) ( )
set )
-valued Function-like quasi_total )
Function of ( ( ) ( non
empty V166()
V167()
V168() )
set ) , ( ( ) ( )
set ) )
. 1 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set )
= p2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) &
g : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V111() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V111() )
TopStruct ) ) : ( ( ) ( non
empty V166()
V167()
V168() )
set )
-defined the
carrier of
((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V118() V119() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non empty V38() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V190() ) ( non empty TopSpace-like V132() V178() V179() V180() V181() V182() V183() V184() V190() ) L15()) | b1 : ( ( ) ( ) Subset of ) ) : ( (
strict ) (
strict TopSpace-like )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V190() ) ( non
empty TopSpace-like V132()
V178()
V179()
V180()
V181()
V182()
V183()
V184()
V190() )
L15()) ) : ( ( ) ( )
set )
-valued Function-like quasi_total )
Function of ( ( ) ( non
empty V166()
V167()
V168() )
set ) , ( ( ) ( )
set ) )
. s1 : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) ( )
set )
= q1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) &
g : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V111() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V111() )
TopStruct ) ) : ( ( ) ( non
empty V166()
V167()
V168() )
set )
-defined the
carrier of
((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V118() V119() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non empty V38() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V190() ) ( non empty TopSpace-like V132() V178() V179() V180() V181() V182() V183() V184() V190() ) L15()) | b1 : ( ( ) ( ) Subset of ) ) : ( (
strict ) (
strict TopSpace-like )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V190() ) ( non
empty TopSpace-like V132()
V178()
V179()
V180()
V181()
V182()
V183()
V184()
V190() )
L15()) ) : ( ( ) ( )
set )
-valued Function-like quasi_total )
Function of ( ( ) ( non
empty V166()
V167()
V168() )
set ) , ( ( ) ( )
set ) )
. s2 : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) ( )
set )
= q2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) &
0 : ( ( ) (
empty V10()
V11()
V12()
ext-real non
positive non
negative Function-like functional V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) )
<= s1 : ( ( ) (
V11()
V12()
ext-real )
Real) &
s1 : ( ( ) (
V11()
V12()
ext-real )
Real)
<= s2 : ( ( ) (
V11()
V12()
ext-real )
Real) &
s2 : ( ( ) (
V11()
V12()
ext-real )
Real)
<= 1 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) ) ;
theorem
for
A being ( ( ) ( )
Subset of )
for
p1,
p2,
q1,
q2 being ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) st
A : ( ( ) ( )
Subset of )
is_an_arc_of p1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) &
LE q1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
q2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
A : ( ( ) ( )
Subset of ) ,
p1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) &
q1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) )
<> q2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) holds
ex
g being ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V111() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V111() )
TopStruct ) ) : ( ( ) ( non
empty V166()
V167()
V168() )
set )
-defined the
carrier of
((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V118() V119() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non empty V38() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V190() ) ( non empty TopSpace-like V132() V178() V179() V180() V181() V182() V183() V184() V190() ) L15()) | b1 : ( ( ) ( ) Subset of ) ) : ( (
strict ) (
strict TopSpace-like )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V190() ) ( non
empty TopSpace-like V132()
V178()
V179()
V180()
V181()
V182()
V183()
V184()
V190() )
L15()) ) : ( ( ) ( )
set )
-valued Function-like quasi_total )
Function of ( ( ) ( non
empty V166()
V167()
V168() )
set ) , ( ( ) ( )
set ) ) ex
s1,
s2 being ( ( ) (
V11()
V12()
ext-real )
Real) st
(
g : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V111() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V111() )
TopStruct ) ) : ( ( ) ( non
empty V166()
V167()
V168() )
set )
-defined the
carrier of
((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V118() V119() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non empty V38() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V190() ) ( non empty TopSpace-like V132() V178() V179() V180() V181() V182() V183() V184() V190() ) L15()) | b1 : ( ( ) ( ) Subset of ) ) : ( (
strict ) (
strict TopSpace-like )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V190() ) ( non
empty TopSpace-like V132()
V178()
V179()
V180()
V181()
V182()
V183()
V184()
V190() )
L15()) ) : ( ( ) ( )
set )
-valued Function-like quasi_total )
Function of ( ( ) ( non
empty V166()
V167()
V168() )
set ) , ( ( ) ( )
set ) ) is
being_homeomorphism &
g : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V111() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V111() )
TopStruct ) ) : ( ( ) ( non
empty V166()
V167()
V168() )
set )
-defined the
carrier of
((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V118() V119() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non empty V38() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V190() ) ( non empty TopSpace-like V132() V178() V179() V180() V181() V182() V183() V184() V190() ) L15()) | b1 : ( ( ) ( ) Subset of ) ) : ( (
strict ) (
strict TopSpace-like )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V190() ) ( non
empty TopSpace-like V132()
V178()
V179()
V180()
V181()
V182()
V183()
V184()
V190() )
L15()) ) : ( ( ) ( )
set )
-valued Function-like quasi_total )
Function of ( ( ) ( non
empty V166()
V167()
V168() )
set ) , ( ( ) ( )
set ) )
. 0 : ( ( ) (
empty V10()
V11()
V12()
ext-real non
positive non
negative Function-like functional V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set )
= p1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) &
g : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V111() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V111() )
TopStruct ) ) : ( ( ) ( non
empty V166()
V167()
V168() )
set )
-defined the
carrier of
((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V118() V119() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non empty V38() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V190() ) ( non empty TopSpace-like V132() V178() V179() V180() V181() V182() V183() V184() V190() ) L15()) | b1 : ( ( ) ( ) Subset of ) ) : ( (
strict ) (
strict TopSpace-like )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V190() ) ( non
empty TopSpace-like V132()
V178()
V179()
V180()
V181()
V182()
V183()
V184()
V190() )
L15()) ) : ( ( ) ( )
set )
-valued Function-like quasi_total )
Function of ( ( ) ( non
empty V166()
V167()
V168() )
set ) , ( ( ) ( )
set ) )
. 1 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set )
= p2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) &
g : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V111() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V111() )
TopStruct ) ) : ( ( ) ( non
empty V166()
V167()
V168() )
set )
-defined the
carrier of
((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V118() V119() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non empty V38() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V190() ) ( non empty TopSpace-like V132() V178() V179() V180() V181() V182() V183() V184() V190() ) L15()) | b1 : ( ( ) ( ) Subset of ) ) : ( (
strict ) (
strict TopSpace-like )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V190() ) ( non
empty TopSpace-like V132()
V178()
V179()
V180()
V181()
V182()
V183()
V184()
V190() )
L15()) ) : ( ( ) ( )
set )
-valued Function-like quasi_total )
Function of ( ( ) ( non
empty V166()
V167()
V168() )
set ) , ( ( ) ( )
set ) )
. s1 : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) ( )
set )
= q1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) &
g : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V111() )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V111() )
TopStruct ) ) : ( ( ) ( non
empty V166()
V167()
V168() )
set )
-defined the
carrier of
((TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V118() V119() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non empty V38() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V190() ) ( non empty TopSpace-like V132() V178() V179() V180() V181() V182() V183() V184() V190() ) L15()) | b1 : ( ( ) ( ) Subset of ) ) : ( (
strict ) (
strict TopSpace-like )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V190() ) ( non
empty TopSpace-like V132()
V178()
V179()
V180()
V181()
V182()
V183()
V184()
V190() )
L15()) ) : ( ( ) ( )
set )
-valued Function-like quasi_total )
Function of ( ( ) ( non
empty V166()
V167()
V168() )
set ) , ( ( ) ( )
set ) )
. s2 : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) ( )
set )
= q2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) &
0 : ( ( ) (
empty V10()
V11()
V12()
ext-real non
positive non
negative Function-like functional V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) )
<= s1 : ( ( ) (
V11()
V12()
ext-real )
Real) &
s1 : ( ( ) (
V11()
V12()
ext-real )
Real)
< s2 : ( ( ) (
V11()
V12()
ext-real )
Real) &
s2 : ( ( ) (
V11()
V12()
ext-real )
Real)
<= 1 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) ) ;
theorem
for
A being ( ( ) ( )
Subset of )
for
q1,
q2,
p1,
p2 being ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) st
LE q1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
q2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
A : ( ( ) ( )
Subset of ) ,
p1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) holds
not
Segment (
A : ( ( ) ( )
Subset of ) ,
p1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
q1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
q2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K6( the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V118() V119() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non empty V38() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V190() ) ( non
empty TopSpace-like V132()
V178()
V179()
V180()
V181()
V182()
V183()
V184()
V190() )
L15()) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) is
empty ;
theorem
for
A being ( ( ) ( )
Subset of )
for
p1,
p2,
q1,
q2 being ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) st
A : ( ( ) ( )
Subset of )
is_an_arc_of p1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) &
LE q1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
q2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
A : ( ( ) ( )
Subset of ) ,
p1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) &
q1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) )
<> q2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) holds
Segment (
A : ( ( ) ( )
Subset of ) ,
p1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
q1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
q2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K6( the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() ext-real positive non negative V118() V119() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non empty V38() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V190() ) ( non
empty TopSpace-like V132()
V178()
V179()
V180()
V181()
V182()
V183()
V184()
V190() )
L15()) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
is_an_arc_of q1 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ,
q2 : ( ( ) (
V45(2 : ( ( ) ( non
empty V10()
V11()
V12()
ext-real positive non
negative V118()
V119()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) (
V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
empty V38()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
V115()
V158() )
Point of ( ( ) ( non
empty )
set ) ) ;