begin
begin
theorem
for
p1,
p2 being ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
for
u1,
u2 being ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) st
u1 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) )
= p1 : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) ) &
u2 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) )
= p2 : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) ) holds
(Pitag_dist 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like V30(
K20(
(REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
functional non
empty FinSequence-membered )
FinSequenceSet of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) ,
(REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
functional non
empty FinSequence-membered )
FinSequenceSet of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) ) : ( ( ) ( )
set ) ,
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) ) (
Relation-like K20(
(REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
functional non
empty FinSequence-membered )
FinSequenceSet of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) ,
(REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
functional non
empty FinSequence-membered )
FinSequenceSet of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) ) : ( ( ) ( )
set )
-defined REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set )
-valued Function-like V30(
K20(
(REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
functional non
empty FinSequence-membered )
FinSequenceSet of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) ,
(REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
functional non
empty FinSequence-membered )
FinSequenceSet of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) ) : ( ( ) ( )
set ) ,
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) )
complex-yielding V128()
V129() )
Element of
K19(
K20(
K20(
(REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
functional non
empty FinSequence-membered )
FinSequenceSet of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) ,
(REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
functional non
empty FinSequence-membered )
FinSequenceSet of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) ) : ( ( ) ( )
set ) ,
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) (
complex-yielding V128()
V129() )
set ) ) : ( ( ) ( )
set ) )
. (
u1 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) ,
u2 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
set )
= sqrt ((((p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) - (p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ^2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) + (((p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) - (p2 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ^2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) ;
theorem
for
p1,
p2,
p being ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
for
r1,
s1,
r2,
s2,
r being ( (
real ) (
V22()
real ext-real )
number )
for
u being ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) st
u : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) )
= p1 : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) ) &
p1 : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
= |[r1 : ( ( real ) ( V22() real ext-real ) number ) ,s1 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) (
Relation-like NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) )
-defined Function-like non
empty V37()
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like FinSubsequence-like complex-yielding V128()
V129() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) ) &
p2 : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
= |[r2 : ( ( real ) ( V22() real ext-real ) number ) ,s2 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) (
Relation-like NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) )
-defined Function-like non
empty V37()
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like FinSubsequence-like complex-yielding V128()
V129() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) ) &
p : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
= |[r2 : ( ( real ) ( V22() real ext-real ) number ) ,s1 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) (
Relation-like NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) )
-defined Function-like non
empty V37()
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like FinSubsequence-like complex-yielding V128()
V129() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) ) &
p2 : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
in Ball (
u : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) ,
r : ( (
real ) (
V22()
real ext-real )
number ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict Reflexive discerning V86()
triangle ) ( non
empty strict Reflexive discerning V86()
triangle )
MetrStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) holds
p : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
in Ball (
u : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) ,
r : ( (
real ) (
V22()
real ext-real )
number ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict Reflexive discerning V86()
triangle ) ( non
empty strict Reflexive discerning V86()
triangle )
MetrStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r1,
r2,
r,
s1,
s2 being ( (
real ) (
V22()
real ext-real )
number )
for
u being ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) st
|[r1 : ( ( real ) ( V22() real ext-real ) number ) ,r2 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) (
Relation-like NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) )
-defined Function-like non
empty V37()
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like FinSubsequence-like complex-yielding V128()
V129() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) )
in Ball (
u : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) ,
r : ( (
real ) (
V22()
real ext-real )
number ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict Reflexive discerning V86()
triangle ) ( non
empty strict Reflexive discerning V86()
triangle )
MetrStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
|[s1 : ( ( real ) ( V22() real ext-real ) number ) ,s2 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) (
Relation-like NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) )
-defined Function-like non
empty V37()
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like FinSubsequence-like complex-yielding V128()
V129() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) )
in Ball (
u : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) ,
r : ( (
real ) (
V22()
real ext-real )
number ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict Reflexive discerning V86()
triangle ) ( non
empty strict Reflexive discerning V86()
triangle )
MetrStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & not
|[r1 : ( ( real ) ( V22() real ext-real ) number ) ,s2 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) (
Relation-like NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) )
-defined Function-like non
empty V37()
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like FinSubsequence-like complex-yielding V128()
V129() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) )
in Ball (
u : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) ,
r : ( (
real ) (
V22()
real ext-real )
number ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict Reflexive discerning V86()
triangle ) ( non
empty strict Reflexive discerning V86()
triangle )
MetrStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) holds
|[s1 : ( ( real ) ( V22() real ext-real ) number ) ,r2 : ( ( real ) ( V22() real ext-real ) number ) ]| : ( ( ) (
Relation-like NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) )
-defined Function-like non
empty V37()
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like FinSubsequence-like complex-yielding V128()
V129() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) )
in Ball (
u : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) ,
r : ( (
real ) (
V22()
real ext-real )
number ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict Reflexive discerning V86()
triangle ) ( non
empty strict Reflexive discerning V86()
triangle )
MetrStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
p1,
p,
q being ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
for
r being ( (
real ) (
V22()
real ext-real )
number )
for
u being ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) st not
p1 : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
in Ball (
u : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) ,
r : ( (
real ) (
V22()
real ext-real )
number ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict Reflexive discerning V86()
triangle ) ( non
empty strict Reflexive discerning V86()
triangle )
MetrStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
p : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
in Ball (
u : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) ,
r : ( (
real ) (
V22()
real ext-real )
number ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict Reflexive discerning V86()
triangle ) ( non
empty strict Reflexive discerning V86()
triangle )
MetrStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
|[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) (
Relation-like NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) )
-defined Function-like non
empty V37()
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like FinSubsequence-like complex-yielding V128()
V129() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) )
in Ball (
u : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) ,
r : ( (
real ) (
V22()
real ext-real )
number ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict Reflexive discerning V86()
triangle ) ( non
empty strict Reflexive discerning V86()
triangle )
MetrStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & not
|[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) (
Relation-like NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) )
-defined Function-like non
empty V37()
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like FinSubsequence-like complex-yielding V128()
V129() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) )
in LSeg (
p1 : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) ) ,
p : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
p1 : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
`1 : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) )
= p : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
`1 : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) &
p : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
`1 : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) )
<> q : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
`1 : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) &
p : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
`2 : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) )
<> q : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
`2 : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) holds
((LSeg (p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,|[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) \/ (LSeg (|[(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
/\ (LSeg (p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( )
Element of
K19( the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
= {p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) ;
theorem
for
p1,
p,
q being ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
for
r being ( (
real ) (
V22()
real ext-real )
number )
for
u being ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) st not
p1 : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
in Ball (
u : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) ,
r : ( (
real ) (
V22()
real ext-real )
number ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict Reflexive discerning V86()
triangle ) ( non
empty strict Reflexive discerning V86()
triangle )
MetrStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
p : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
in Ball (
u : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) ,
r : ( (
real ) (
V22()
real ext-real )
number ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict Reflexive discerning V86()
triangle ) ( non
empty strict Reflexive discerning V86()
triangle )
MetrStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
|[(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) (
Relation-like NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) )
-defined Function-like non
empty V37()
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like FinSubsequence-like complex-yielding V128()
V129() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) )
in Ball (
u : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) ,
r : ( (
real ) (
V22()
real ext-real )
number ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(Euclid 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict Reflexive discerning V86()
triangle ) ( non
empty strict Reflexive discerning V86()
triangle )
MetrStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & not
|[(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) (
Relation-like NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) )
-defined Function-like non
empty V37()
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like FinSubsequence-like complex-yielding V128()
V129() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) )
in LSeg (
p1 : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) ) ,
p : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) &
p1 : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
`2 : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) )
= p : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
`2 : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) &
p : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
`1 : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) )
<> q : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
`1 : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) &
p : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
`2 : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) )
<> q : ( ( ) (
V44(2 : ( ( ) ( non
empty ordinal natural V22()
real ext-real positive non
negative V33()
V34()
V137()
V138()
V139()
V140()
V141()
V142() )
Element of
NAT : ( ( ) (
V137()
V138()
V139()
V140()
V141()
V142()
V143() )
Element of
K19(
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like V129() )
Point of ( ( ) ( non
empty )
set ) )
`2 : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V37()
V137()
V138()
V139()
V143() )
set ) ) holds
((LSeg (p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,|[(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) \/ (LSeg (|[(q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `1) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ,(p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) `2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) ]| : ( ( ) ( Relation-like NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like non empty V37() V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-yielding V128() V129() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of K19( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like V103() V149() V150() V151() V152() V153() V154() V155() strict ) RLTopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
/\ (LSeg (p1 : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( )
Element of
K19( the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19( the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
strict ) ( non
empty TopSpace-like V103()
V149()
V150()
V151()
V152()
V153()
V154()
V155()
strict )
RLTopStruct ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
= {p : ( ( ) ( V44(2 : ( ( ) ( non empty ordinal natural V22() real ext-real positive non negative V33() V34() V137() V138() V139() V140() V141() V142() ) Element of NAT : ( ( ) ( V137() V138() V139() V140() V141() V142() V143() ) Element of K19(REAL : ( ( ) ( non empty V37() V137() V138() V139() V143() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V129() ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non
empty )
set ) ;