begin
definition
let a,
b,
c,
x,
y,
z be ( ( ) ( )
set ) ;
func (
a,
b,
c)
--> (
x,
y,
z)
-> ( ( ) ( )
set )
equals
((a : ( ( non empty ) ( non empty ) TopStruct ) ,b : ( ( ) ( ) Element of the carrier of a : ( ( non empty ) ( non empty ) TopStruct ) : ( ( ) ( non empty ) set ) ) ) --> (x : ( ( Function-like quasi_total ) ( Relation-like [:REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) ,a : ( ( non empty ) ( non empty ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) -defined a : ( ( non empty ) ( non empty ) TopStruct ) -valued Function-like quasi_total ) Element of bool [:[:REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) ,a : ( ( non empty ) ( non empty ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) ,a : ( ( non empty ) ( non empty ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) ,y : ( ( ) ( ) Element of bool (bool a : ( ( non empty ) ( non empty ) TopStruct ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) )) : ( ( ) (
Relation-like Function-like )
set )
+* (c : ( ( Function-like quasi_total ) ( Relation-like [:a : ( ( non empty ) ( non empty ) TopStruct ) ,a : ( ( non empty ) ( non empty ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) -defined a : ( ( non empty ) ( non empty ) TopStruct ) -valued Function-like quasi_total ) Element of bool [:[:a : ( ( non empty ) ( non empty ) TopStruct ) ,a : ( ( non empty ) ( non empty ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) ,a : ( ( non empty ) ( non empty ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) .--> z : ( ( ) ( ) Element of the carrier of b : ( ( ) ( ) Element of the carrier of a : ( ( non empty ) ( non empty ) TopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
Relation-like {c : ( ( Function-like quasi_total ) ( Relation-like [:a : ( ( non empty ) ( non empty ) TopStruct ) ,a : ( ( non empty ) ( non empty ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) -defined a : ( ( non empty ) ( non empty ) TopStruct ) -valued Function-like quasi_total ) Element of bool [:[:a : ( ( non empty ) ( non empty ) TopStruct ) ,a : ( ( non empty ) ( non empty ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) ,a : ( ( non empty ) ( non empty ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) } : ( ( ) (
functional non
empty )
set )
-defined Function-like one-to-one constant trivial )
set ) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set ) ;
end;
registration
let a,
b,
c,
x,
y,
z be ( ( ) ( )
set ) ;
end;
theorem
for
a,
b,
x,
y,
z being ( ( ) ( )
set )
for
c being ( (
complex ) (
complex )
number ) holds
dom ((a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ,c : ( ( complex ) ( complex ) number ) ) --> (x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ,z : ( ( ) ( ) set ) )) : ( ( ) (
Relation-like Function-like )
set ) : ( ( ) ( )
set )
= {a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ,c : ( ( complex ) ( complex ) number ) } : ( ( ) ( )
set ) ;
theorem
for
a,
b,
x,
y,
z being ( ( ) ( )
set )
for
c being ( (
complex ) (
complex )
number ) holds
rng ((a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ,c : ( ( complex ) ( complex ) number ) ) --> (x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ,z : ( ( ) ( ) set ) )) : ( ( ) (
Relation-like Function-like )
set ) : ( ( ) ( )
set )
c= {x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ,z : ( ( ) ( ) set ) } : ( ( ) ( )
set ) ;
theorem
for
a,
b,
x,
y,
z being ( ( ) ( )
set ) holds (
a : ( ( ) ( )
set ) ,
a : ( ( ) ( )
set ) ,
b : ( ( ) ( )
set ) )
--> (
x : ( ( ) ( )
set ) ,
y : ( ( ) ( )
set ) ,
z : ( ( ) ( )
set ) ) : ( ( ) (
Relation-like Function-like )
set )
= (
a : ( ( ) ( )
set ) ,
b : ( ( ) ( )
set ) )
--> (
y : ( ( ) ( )
set ) ,
z : ( ( ) ( )
set ) ) : ( ( ) (
Relation-like Function-like )
set ) ;
theorem
for
a,
b,
x,
y,
z being ( ( ) ( )
set ) st
a : ( ( ) ( )
set )
<> b : ( ( ) ( )
set ) holds
(
a : ( ( ) ( )
set ) ,
b : ( ( ) ( )
set ) ,
a : ( ( ) ( )
set ) )
--> (
x : ( ( ) ( )
set ) ,
y : ( ( ) ( )
set ) ,
z : ( ( ) ( )
set ) ) : ( ( ) (
Relation-like Function-like )
set )
= (
a : ( ( ) ( )
set ) ,
b : ( ( ) ( )
set ) )
--> (
z : ( ( ) ( )
set ) ,
y : ( ( ) ( )
set ) ) : ( ( ) (
Relation-like Function-like )
set ) ;
theorem
for
a,
b,
x,
y,
z being ( ( ) ( )
set ) holds (
a : ( ( ) ( )
set ) ,
b : ( ( ) ( )
set ) ,
b : ( ( ) ( )
set ) )
--> (
x : ( ( ) ( )
set ) ,
y : ( ( ) ( )
set ) ,
z : ( ( ) ( )
set ) ) : ( ( ) (
Relation-like Function-like )
set )
= (
a : ( ( ) ( )
set ) ,
b : ( ( ) ( )
set ) )
--> (
x : ( ( ) ( )
set ) ,
z : ( ( ) ( )
set ) ) : ( ( ) (
Relation-like Function-like )
set ) ;
theorem
for
a,
b,
x,
y,
z being ( ( ) ( )
set )
for
c being ( (
complex ) (
complex )
number ) st
a : ( ( ) ( )
set ) ,
b : ( ( ) ( )
set ) ,
c : ( (
complex ) (
complex )
number )
are_mutually_different holds
(
((a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ,c : ( ( complex ) ( complex ) number ) ) --> (x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ,z : ( ( ) ( ) set ) )) : ( ( ) (
Relation-like Function-like )
set )
. a : ( ( ) ( )
set ) : ( ( ) ( )
set )
= x : ( ( ) ( )
set ) &
((a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ,c : ( ( complex ) ( complex ) number ) ) --> (x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ,z : ( ( ) ( ) set ) )) : ( ( ) (
Relation-like Function-like )
set )
. b : ( ( ) ( )
set ) : ( ( ) ( )
set )
= y : ( ( ) ( )
set ) &
((a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ,c : ( ( complex ) ( complex ) number ) ) --> (x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ,z : ( ( ) ( ) set ) )) : ( ( ) (
Relation-like Function-like )
set )
. c : ( (
complex ) (
complex )
number ) : ( ( ) ( )
set )
= z : ( ( ) ( )
set ) ) ;
theorem
for
a,
b,
x,
y,
z being ( ( ) ( )
set )
for
c being ( (
complex ) (
complex )
number )
for
f being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) st
dom f : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) : ( ( ) ( )
set )
= {a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ,c : ( ( complex ) ( complex ) number ) } : ( ( ) ( )
set ) &
f : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. a : ( ( ) ( )
set ) : ( ( ) ( )
set )
= x : ( ( ) ( )
set ) &
f : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. b : ( ( ) ( )
set ) : ( ( ) ( )
set )
= y : ( ( ) ( )
set ) &
f : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. c : ( (
complex ) (
complex )
number ) : ( ( ) ( )
set )
= z : ( ( ) ( )
set ) holds
f : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
= (
a : ( ( ) ( )
set ) ,
b : ( ( ) ( )
set ) ,
c : ( (
complex ) (
complex )
number ) )
--> (
x : ( ( ) ( )
set ) ,
y : ( ( ) ( )
set ) ,
z : ( ( ) ( )
set ) ) : ( ( ) (
Relation-like Function-like )
set ) ;
theorem
for
a,
b,
x,
y,
z being ( ( ) ( )
set )
for
c being ( (
complex ) (
complex )
number ) st
a : ( ( ) ( )
set ) ,
b : ( ( ) ( )
set ) ,
c : ( (
complex ) (
complex )
number )
are_mutually_different holds
product ((a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ,c : ( ( complex ) ( complex ) number ) ) --> ({x : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) ,{y : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) ,{z : ( ( ) ( ) set ) } : ( ( ) ( non empty ) set ) )) : ( ( ) (
Relation-like Function-like )
set ) : ( ( ) ( )
set )
= {((a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ,c : ( ( complex ) ( complex ) number ) ) --> (x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ,z : ( ( ) ( ) set ) )) : ( ( ) ( Relation-like Function-like ) set ) } : ( ( ) (
functional non
empty )
set ) ;
theorem
for
a,
b being ( ( ) ( )
set )
for
c being ( (
complex ) (
complex )
number )
for
A,
B,
C,
D,
E,
F being ( ( ) ( )
set ) st
A : ( ( ) ( )
set )
c= B : ( ( ) ( )
set ) &
C : ( ( ) ( )
set )
c= D : ( ( ) ( )
set ) &
E : ( ( ) ( )
set )
c= F : ( ( ) ( )
set ) holds
product ((a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ,c : ( ( complex ) ( complex ) number ) ) --> (A : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ,E : ( ( ) ( ) set ) )) : ( ( ) (
Relation-like Function-like )
set ) : ( ( ) ( )
set )
c= product ((a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ,c : ( ( complex ) ( complex ) number ) ) --> (B : ( ( ) ( ) set ) ,D : ( ( ) ( ) set ) ,F : ( ( ) ( ) set ) )) : ( ( ) (
Relation-like Function-like )
set ) : ( ( ) ( )
set ) ;
theorem
for
a,
b,
x,
X,
y,
Y,
z,
Z being ( ( ) ( )
set )
for
c being ( (
complex ) (
complex )
number ) st
a : ( ( ) ( )
set ) ,
b : ( ( ) ( )
set ) ,
c : ( (
complex ) (
complex )
number )
are_mutually_different &
x : ( ( ) ( )
set )
in X : ( ( ) ( )
set ) &
y : ( ( ) ( )
set )
in Y : ( ( ) ( )
set ) &
z : ( ( ) ( )
set )
in Z : ( ( ) ( )
set ) holds
(
a : ( ( ) ( )
set ) ,
b : ( ( ) ( )
set ) ,
c : ( (
complex ) (
complex )
number ) )
--> (
x : ( ( ) ( )
set ) ,
y : ( ( ) ( )
set ) ,
z : ( ( ) ( )
set ) ) : ( ( ) (
Relation-like Function-like )
set )
in product ((a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ,c : ( ( complex ) ( complex ) number ) ) --> (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ,Z : ( ( ) ( ) set ) )) : ( ( ) (
Relation-like Function-like )
set ) : ( ( ) ( )
set ) ;
begin
definition
let x0,
y0 be ( ( ) (
Relation-like Function-like complex-yielding ext-real-valued real-valued )
Point of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) ) ;
let xt be ( ( ) ( )
set ) ;
let f be ( ( ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V224()
V231()
pathwise_connected connected )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V231()
pathwise_connected connected )
TopStruct ) ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-defined the
carrier of
(Tunit_circle 2 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty TopSpace-like V241()
complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like non
empty total quasi_total Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Path of
x0 : ( ( ) (
Relation-like Function-like complex-yielding ext-real-valued real-valued )
Point of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) ) ,
y0 : ( ( ) (
Relation-like Function-like complex-yielding ext-real-valued real-valued )
Point of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) ) ) ;
assume
xt : ( ( ) ( )
set )
in CircleMap : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
K708() : ( (
V240() ) ( non
empty strict TopSpace-like V231()
pathwise_connected V240()
connected )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V231()
pathwise_connected connected )
TopStruct ) ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-defined the
carrier of
(Tunit_circle 2 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty TopSpace-like V241()
complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like non
empty total quasi_total onto continuous Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Element of
bool [: the carrier of K708() : ( ( V240() ) ( non empty strict TopSpace-like V231() pathwise_connected V240() connected ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V231() pathwise_connected connected ) TopStruct ) ) : ( ( ) ( non empty V195() V196() V197() ) set ) , the carrier of (Tunit_circle 2 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like V241() complex-functions-membered real-functions-membered ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
" {x0 : ( ( ) ( Relation-like Function-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) } : ( ( ) (
functional non
empty complex-functions-membered )
set ) : ( ( ) (
V195()
V196()
V197() )
Element of
bool the
carrier of
K708() : ( (
V240() ) ( non
empty strict TopSpace-like V231()
pathwise_connected V240()
connected )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V231()
pathwise_connected connected )
TopStruct ) ) : ( ( ) ( non
empty V195()
V196()
V197() )
set ) : ( ( ) (
V74() )
set ) )
;
func liftPath (
f,
xt)
-> ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V224()
V231()
pathwise_connected connected )
TopStruct ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-defined the
carrier of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V231()
pathwise_connected connected )
TopStruct ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-valued Function-like non
empty total quasi_total complex-yielding ext-real-valued real-valued )
Function of ( ( ) ( non
empty V195()
V196()
V197() )
set ) , ( ( ) ( non
empty V195()
V196()
V197() )
set ) )
means
(
it : ( ( ) ( )
set )
. 0 : ( ( ) (
Relation-like non-empty empty-yielding NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) )
-defined RAT : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V198()
V201() )
set )
-valued Function-like one-to-one constant functional empty complex ordinal natural real integer non
irrational V32()
FinSequence-like FinSubsequence-like FinSequence-membered ext-real non
positive non
negative complex-yielding ext-real-valued real-valued natural-valued Function-yielding V178()
V195()
V196()
V197()
V198()
V199()
V200()
V201()
complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued odd )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( ( ) ( )
set )
= xt : ( (
real positive ) ( non
empty complex real ext-real positive non
negative )
set ) &
f : ( ( ) ( )
set )
= CircleMap : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
K708() : ( (
V240() ) ( non
empty strict TopSpace-like V231()
pathwise_connected V240()
connected )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V231()
pathwise_connected connected )
TopStruct ) ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-defined the
carrier of
(Tunit_circle 2 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty TopSpace-like V241()
complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like non
empty total quasi_total onto continuous Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Element of
bool [: the carrier of K708() : ( ( V240() ) ( non empty strict TopSpace-like V231() pathwise_connected V240() connected ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V231() pathwise_connected connected ) TopStruct ) ) : ( ( ) ( non empty V195() V196() V197() ) set ) , the carrier of (Tunit_circle 2 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like V241() complex-functions-membered real-functions-membered ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
* it : ( ( ) ( )
set ) : ( (
Function-like ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V224()
V231()
pathwise_connected connected )
TopStruct ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-defined the
carrier of
(Tunit_circle 2 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty TopSpace-like V241()
complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Element of
bool [: the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V224() V231() pathwise_connected connected ) TopStruct ) : ( ( ) ( non empty V195() V196() V197() ) set ) , the carrier of (Tunit_circle 2 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like V241() complex-functions-membered real-functions-membered ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) &
it : ( ( ) ( )
set ) is
continuous & ( for
f1 being ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V224()
V231()
pathwise_connected connected )
TopStruct ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-defined the
carrier of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V231()
pathwise_connected connected )
TopStruct ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-valued Function-like non
empty total quasi_total complex-yielding ext-real-valued real-valued )
Function of ( ( ) ( non
empty V195()
V196()
V197() )
set ) , ( ( ) ( non
empty V195()
V196()
V197() )
set ) ) st
f1 : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V224()
V231()
pathwise_connected connected )
TopStruct ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-defined the
carrier of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V231()
pathwise_connected connected )
TopStruct ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-valued Function-like non
empty total quasi_total complex-yielding ext-real-valued real-valued )
Function of ( ( ) ( non
empty V195()
V196()
V197() )
set ) , ( ( ) ( non
empty V195()
V196()
V197() )
set ) ) is
continuous &
f : ( ( ) ( )
set )
= CircleMap : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
K708() : ( (
V240() ) ( non
empty strict TopSpace-like V231()
pathwise_connected V240()
connected )
SubSpace of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V231()
pathwise_connected connected )
TopStruct ) ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-defined the
carrier of
(Tunit_circle 2 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty TopSpace-like V241()
complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like non
empty total quasi_total onto continuous Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Element of
bool [: the carrier of K708() : ( ( V240() ) ( non empty strict TopSpace-like V231() pathwise_connected V240() connected ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V231() pathwise_connected connected ) TopStruct ) ) : ( ( ) ( non empty V195() V196() V197() ) set ) , the carrier of (Tunit_circle 2 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like V241() complex-functions-membered real-functions-membered ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
* f1 : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V224()
V231()
pathwise_connected connected )
TopStruct ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-defined the
carrier of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V231()
pathwise_connected connected )
TopStruct ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-valued Function-like non
empty total quasi_total complex-yielding ext-real-valued real-valued )
Function of ( ( ) ( non
empty V195()
V196()
V197() )
set ) , ( ( ) ( non
empty V195()
V196()
V197() )
set ) ) : ( (
Function-like ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V224()
V231()
pathwise_connected connected )
TopStruct ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-defined the
carrier of
(Tunit_circle 2 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty TopSpace-like V241()
complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL 2 : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like non
empty total quasi_total Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Element of
bool [: the carrier of I[01] : ( ( ) ( non empty strict TopSpace-like V224() V231() pathwise_connected connected ) TopStruct ) : ( ( ) ( non empty V195() V196() V197() ) set ) , the carrier of (Tunit_circle 2 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like V241() complex-functions-membered real-functions-membered ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) &
f1 : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V224()
V231()
pathwise_connected connected )
TopStruct ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-defined the
carrier of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V231()
pathwise_connected connected )
TopStruct ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-valued Function-like non
empty total quasi_total complex-yielding ext-real-valued real-valued )
Function of ( ( ) ( non
empty V195()
V196()
V197() )
set ) , ( ( ) ( non
empty V195()
V196()
V197() )
set ) )
. 0 : ( ( ) (
Relation-like non-empty empty-yielding NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) )
-defined RAT : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V198()
V201() )
set )
-valued Function-like one-to-one constant functional empty complex ordinal natural real integer non
irrational V32()
FinSequence-like FinSubsequence-like FinSequence-membered ext-real non
positive non
negative complex-yielding ext-real-valued real-valued natural-valued Function-yielding V178()
V195()
V196()
V197()
V198()
V199()
V200()
V201()
complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued odd )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( ( ) (
complex real ext-real )
set )
= xt : ( (
real positive ) ( non
empty complex real ext-real positive non
negative )
set ) holds
it : ( ( ) ( )
set )
= f1 : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
I[01] : ( ( ) ( non
empty strict TopSpace-like V224()
V231()
pathwise_connected connected )
TopStruct ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-defined the
carrier of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V231()
pathwise_connected connected )
TopStruct ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-valued Function-like non
empty total quasi_total complex-yielding ext-real-valued real-valued )
Function of ( ( ) ( non
empty V195()
V196()
V197() )
set ) , ( ( ) ( non
empty V195()
V196()
V197() )
set ) ) ) );
end;
theorem
for
n being ( ( non
empty natural ) ( non
empty complex ordinal natural real integer ext-real positive non
negative )
Nat)
for
p,
x,
y,
x1,
y1 being ( ( ) (
Relation-like NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) )
-defined Function-like V32()
V39(
b1 : ( ( non
empty natural ) ( non
empty complex ordinal natural real integer ext-real positive non
negative )
Nat) )
FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued )
Point of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) )
for
r being ( (
real positive ) ( non
empty complex real ext-real positive non
negative )
number ) st
x : ( ( ) (
Relation-like NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) )
-defined Function-like V32()
V39(
b1 : ( ( non
empty natural ) ( non
empty complex ordinal natural real integer ext-real positive non
negative )
Nat) )
FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued )
Point of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) ) ,
y : ( ( ) (
Relation-like NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) )
-defined Function-like V32()
V39(
b1 : ( ( non
empty natural ) ( non
empty complex ordinal natural real integer ext-real positive non
negative )
Nat) )
FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued )
Point of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) )
are_antipodals_of 0. (TOP-REAL n : ( ( non empty natural ) ( non empty complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) : ( ( ) (
Relation-like NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) )
-defined Function-like V32()
V39(
b1 : ( ( non
empty natural ) ( non
empty complex ordinal natural real integer ext-real positive non
negative )
Nat) )
zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued )
Element of the
carrier of
(TOP-REAL b1 : ( ( non empty natural ) ( non empty complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) ) ,1 : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) &
x1 : ( ( ) (
Relation-like NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) )
-defined Function-like V32()
V39(
b1 : ( ( non
empty natural ) ( non
empty complex ordinal natural real integer ext-real positive non
negative )
Nat) )
FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued )
Point of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) )
= (CircleIso (p : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non empty natural ) ( non empty complex ordinal natural real integer ext-real positive non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,r : ( ( real positive ) ( non empty complex real ext-real positive non negative ) number ) )) : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
(Tunit_circle b1 : ( ( non empty natural ) ( non empty complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( ) ( non
empty TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL b1 : ( ( non
empty natural ) ( non
empty complex ordinal natural real integer ext-real positive non
negative )
Nat) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-defined the
carrier of
(Tcircle (b2 : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non empty natural ) ( non empty complex ordinal natural real integer ext-real positive non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,b7 : ( ( real positive ) ( non empty complex real ext-real positive non negative ) number ) )) : ( ( ) ( non
empty strict TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL b1 : ( ( non
empty natural ) ( non
empty complex ordinal natural real integer ext-real positive non
negative )
Nat) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like one-to-one non
empty total quasi_total onto bijective continuous Function-yielding V178()
being_homeomorphism complex-functions-valued ext-real-functions-valued real-functions-valued )
Function of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) , ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) )
. x : ( ( ) (
Relation-like NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) )
-defined Function-like V32()
V39(
b1 : ( ( non
empty natural ) ( non
empty complex ordinal natural real integer ext-real positive non
negative )
Nat) )
FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued )
Point of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) ) : ( ( ) (
Relation-like Function-like complex-yielding ext-real-valued real-valued )
set ) &
y1 : ( ( ) (
Relation-like NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) )
-defined Function-like V32()
V39(
b1 : ( ( non
empty natural ) ( non
empty complex ordinal natural real integer ext-real positive non
negative )
Nat) )
FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued )
Point of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) )
= (CircleIso (p : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non empty natural ) ( non empty complex ordinal natural real integer ext-real positive non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,r : ( ( real positive ) ( non empty complex real ext-real positive non negative ) number ) )) : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
(Tunit_circle b1 : ( ( non empty natural ) ( non empty complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( ) ( non
empty TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL b1 : ( ( non
empty natural ) ( non
empty complex ordinal natural real integer ext-real positive non
negative )
Nat) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-defined the
carrier of
(Tcircle (b2 : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non empty natural ) ( non empty complex ordinal natural real integer ext-real positive non negative ) Nat) ) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Point of ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,b7 : ( ( real positive ) ( non empty complex real ext-real positive non negative ) number ) )) : ( ( ) ( non
empty strict TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL b1 : ( ( non
empty natural ) ( non
empty complex ordinal natural real integer ext-real positive non
negative )
Nat) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like one-to-one non
empty total quasi_total onto bijective continuous Function-yielding V178()
being_homeomorphism complex-functions-valued ext-real-functions-valued real-functions-valued )
Function of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) , ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) )
. y : ( ( ) (
Relation-like NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) )
-defined Function-like V32()
V39(
b1 : ( ( non
empty natural ) ( non
empty complex ordinal natural real integer ext-real positive non
negative )
Nat) )
FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued )
Point of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) ) : ( ( ) (
Relation-like Function-like complex-yielding ext-real-valued real-valued )
set ) holds
x1 : ( ( ) (
Relation-like NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) )
-defined Function-like V32()
V39(
b1 : ( ( non
empty natural ) ( non
empty complex ordinal natural real integer ext-real positive non
negative )
Nat) )
FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued )
Point of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) ) ,
y1 : ( ( ) (
Relation-like NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) )
-defined Function-like V32()
V39(
b1 : ( ( non
empty natural ) ( non
empty complex ordinal natural real integer ext-real positive non
negative )
Nat) )
FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued )
Point of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) )
are_antipodals_of p : ( ( ) (
Relation-like NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) )
-defined Function-like V32()
V39(
b1 : ( ( non
empty natural ) ( non
empty complex ordinal natural real integer ext-real positive non
negative )
Nat) )
FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued )
Point of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) ) ,
r : ( (
real positive ) ( non
empty complex real ext-real positive non
negative )
number ) ;
theorem
for
n being ( ( non
zero natural ) ( non
zero complex ordinal natural real integer ext-real positive non
negative )
Nat)
for
f,
g,
B1 being ( (
Function-like quasi_total ) (
Relation-like the
carrier of
(Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non
empty strict TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-defined the
carrier of
(TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like non
empty total quasi_total Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Function of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) , ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) ) st
g : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
(Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non
empty strict TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-defined the
carrier of
(TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like non
empty total quasi_total Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Function of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) , ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) )
= f : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
(Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non
empty strict TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-defined the
carrier of
(TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like non
empty total quasi_total Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Function of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) , ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) )
(-) : ( (
Relation-like Function-like ) (
Relation-like Function-like )
set ) &
B1 : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
(Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non
empty strict TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-defined the
carrier of
(TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like non
empty total quasi_total Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Function of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) , ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) )
= f : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
(Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non
empty strict TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-defined the
carrier of
(TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like non
empty total quasi_total Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Function of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) , ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) )
<--> g : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
(Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non
empty strict TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-defined the
carrier of
(TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like non
empty total quasi_total Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Function of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) , ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) ) : ( (
Function-like ) (
Relation-like the
carrier of
(Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non
empty strict TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
/\ the
carrier of
(Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non
empty strict TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) : ( ( ) ( )
set )
-defined R_PFuncs ((DOMS the carrier of (TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS the carrier of (TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( )
set ) : ( ( ) (
functional complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Element of
bool [:( the carrier of (Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non empty strict TopSpace-like complex-functions-membered real-functions-membered ) SubSpace of TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) /\ the carrier of (Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non empty strict TopSpace-like complex-functions-membered real-functions-membered ) SubSpace of TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) : ( ( ) ( ) set ) ,(R_PFuncs ((DOMS the carrier of (TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) : ( ( ) ( ) set ) /\ (DOMS the carrier of (TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) &
f : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
(Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non
empty strict TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-defined the
carrier of
(TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like non
empty total quasi_total Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Function of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) , ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) ) is
without_antipodals holds
Sn1->Sn f : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
(Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non
empty strict TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-defined the
carrier of
(TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like non
empty total quasi_total Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Function of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) , ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) ) : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
(Tunit_circle (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-defined the
carrier of
(Tunit_circle b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( ) ( non
empty TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL b1 : ( ( non
zero natural ) ( non
zero complex ordinal natural real integer ext-real positive non
negative )
Nat) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like non
empty total quasi_total Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Function of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) , ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) )
= B1 : ( (
Function-like quasi_total ) (
Relation-like the
carrier of
(Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non
empty strict TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-defined the
carrier of
(TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like non
empty total quasi_total Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Function of ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) , ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set ) )
</> ((n : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) NormF) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -defined the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V231() pathwise_connected connected ) TopStruct ) : ( ( ) ( non empty V195() V196() V197() ) set ) -valued Function-like non empty total quasi_total continuous complex-yielding ext-real-valued real-valued ) Element of bool [: the carrier of (TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V231() pathwise_connected connected ) TopStruct ) : ( ( ) ( non empty V195() V196() V197() ) set ) :] : ( ( ) ( Relation-like complex-yielding ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) * B1 : ( ( Function-like quasi_total ) ( Relation-like the carrier of (Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non empty strict TopSpace-like complex-functions-membered real-functions-membered ) SubSpace of TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -defined the carrier of (TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) , ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ) : ( (
Function-like ) (
Relation-like the
carrier of
(Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non
empty strict TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-defined the
carrier of
R^1 : ( (
TopSpace-like ) ( non
empty strict TopSpace-like V231()
pathwise_connected connected )
TopStruct ) : ( ( ) ( non
empty V195()
V196()
V197() )
set )
-valued Function-like non
empty total quasi_total complex-yielding ext-real-valued real-valued )
Element of
bool [: the carrier of (Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non empty strict TopSpace-like complex-functions-membered real-functions-membered ) SubSpace of TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V231() pathwise_connected connected ) TopStruct ) : ( ( ) ( non empty V195() V196() V197() ) set ) :] : ( ( ) (
Relation-like complex-yielding ext-real-valued real-valued )
set ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like the
carrier of
(Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non
empty strict TopSpace-like complex-functions-membered real-functions-membered )
SubSpace of
TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non
empty complex ordinal natural real integer non
irrational ext-real positive non
negative V195()
V196()
V197()
V198()
V199()
V200() )
Element of
NAT : ( ( ) (
V195()
V196()
V197()
V198()
V199()
V200()
V201() )
Element of
bool REAL : ( ( ) ( non
empty V32()
V195()
V196()
V197()
V201() )
set ) : ( ( ) (
V74() )
set ) ) ) : ( (
strict ) ( non
empty non
trivial V47()
TopSpace-like V89()
V117()
V118()
V119()
V120()
V121()
V122()
V123()
strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group )
RLTopStruct ) ) : ( ( ) (
functional non
empty complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
/\ (dom ((b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) NormF) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -defined the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V231() pathwise_connected connected ) TopStruct ) : ( ( ) ( non empty V195() V196() V197() ) set ) -valued Function-like non empty total quasi_total continuous complex-yielding ext-real-valued real-valued ) Element of bool [: the carrier of (TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V231() pathwise_connected connected ) TopStruct ) : ( ( ) ( non empty V195() V196() V197() ) set ) :] : ( ( ) ( Relation-like complex-yielding ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) * b4 : ( ( Function-like quasi_total ) ( Relation-like the carrier of (Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non empty strict TopSpace-like complex-functions-membered real-functions-membered ) SubSpace of TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -defined the carrier of (TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) , ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ) : ( ( Function-like ) ( Relation-like the carrier of (Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non empty strict TopSpace-like complex-functions-membered real-functions-membered ) SubSpace of TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -defined the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V231() pathwise_connected connected ) TopStruct ) : ( ( ) ( non empty V195() V196() V197() ) set ) -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued ) Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non empty strict TopSpace-like complex-functions-membered real-functions-membered ) SubSpace of TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V231() pathwise_connected connected ) TopStruct ) : ( ( ) ( non empty V195() V196() V197() ) set ) :] : ( ( ) ( Relation-like complex-yielding ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non
empty )
set ) : ( ( ) ( )
set )
-defined R_PFuncs (DOMS the carrier of (TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) : ( ( ) ( )
set ) : ( ( ) (
functional complex-functions-membered ext-real-functions-membered real-functions-membered )
set )
-valued Function-like Function-yielding V178()
complex-functions-valued ext-real-functions-valued real-functions-valued )
Element of
bool [:( the carrier of (Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non empty strict TopSpace-like complex-functions-membered real-functions-membered ) SubSpace of TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) /\ (dom ((b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) NormF) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -defined the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V231() pathwise_connected connected ) TopStruct ) : ( ( ) ( non empty V195() V196() V197() ) set ) -valued Function-like non empty total quasi_total continuous complex-yielding ext-real-valued real-valued ) Element of bool [: the carrier of (TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V231() pathwise_connected connected ) TopStruct ) : ( ( ) ( non empty V195() V196() V197() ) set ) :] : ( ( ) ( Relation-like complex-yielding ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) * b4 : ( ( Function-like quasi_total ) ( Relation-like the carrier of (Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non empty strict TopSpace-like complex-functions-membered real-functions-membered ) SubSpace of TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -defined the carrier of (TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -valued Function-like non empty total quasi_total Function-yielding V178() complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) , ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ) : ( ( Function-like ) ( Relation-like the carrier of (Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non empty strict TopSpace-like complex-functions-membered real-functions-membered ) SubSpace of TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) -defined the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V231() pathwise_connected connected ) TopStruct ) : ( ( ) ( non empty V195() V196() V197() ) set ) -valued Function-like non empty total quasi_total complex-yielding ext-real-valued real-valued ) Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) -defined Function-like V32() V39(b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) zero FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued ) Element of the carrier of (TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty non trivial V32() complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) ,1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) )) : ( ( ) ( non empty strict TopSpace-like complex-functions-membered real-functions-membered ) SubSpace of TOP-REAL (b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) ) : ( ( ) ( non empty complex ordinal natural real integer non irrational ext-real positive non negative V195() V196() V197() V198() V199() V200() ) Element of NAT : ( ( ) ( V195() V196() V197() V198() V199() V200() V201() ) Element of bool REAL : ( ( ) ( non empty V32() V195() V196() V197() V201() ) set ) : ( ( ) ( V74() ) set ) ) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) , the carrier of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V231() pathwise_connected connected ) TopStruct ) : ( ( ) ( non empty V195() V196() V197() ) set ) :] : ( ( ) ( Relation-like complex-yielding ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ,(R_PFuncs (DOMS the carrier of (TOP-REAL b1 : ( ( non zero natural ) ( non zero complex ordinal natural real integer ext-real positive non negative ) Nat) ) : ( ( strict ) ( non empty non trivial V47() TopSpace-like V89() V117() V118() V119() V120() V121() V122() V123() strict complex-functions-membered real-functions-membered having_trivial_Fundamental_Group ) RLTopStruct ) : ( ( ) ( functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ;