:: TOPREALB semantic presentation

begin

registration
cluster K108(0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty ) set ) -> non empty ;
cluster K105((- 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real non positive negative integer ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty ) set ) -> non empty ;
cluster K108((1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,(3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( ) set ) -> non empty ;
end;

registration
cluster sin : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) -> Function-like quasi_total continuous ;
cluster cos : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) -> Function-like quasi_total continuous ;
cluster arcsin : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like one-to-one V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) -> Function-like continuous ;
cluster arccos : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like one-to-one V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) -> Function-like continuous ;
end;

theorem :: TOPREALB:1
for a, r, b being ( ( real ) ( complex real ext-real ) number ) holds sin ((a : ( ( real ) ( complex real ext-real ) number ) * r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) + b : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) : ( ( ) ( complex real ext-real ) set ) = (sin : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) * (AffineMap (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) )) : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) . r : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( complex real ext-real ) set ) ;

theorem :: TOPREALB:2
for a, r, b being ( ( real ) ( complex real ext-real ) number ) holds cos ((a : ( ( real ) ( complex real ext-real ) number ) * r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) + b : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) : ( ( ) ( complex real ext-real ) set ) = (cos : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) * (AffineMap (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) )) : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) . r : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( complex real ext-real ) set ) ;

registration
let a be ( ( non zero real ) ( non zero complex real ext-real ) number ) ;
let b be ( ( real ) ( complex real ext-real ) number ) ;
cluster AffineMap (a : ( ( non zero real ) ( non zero complex real ext-real ) set ) ,b : ( ( real ) ( complex real ext-real ) set ) ) : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) -> Function-like one-to-one quasi_total onto ;
end;

definition
let a, b be ( ( real ) ( complex real ext-real ) number ) ;
func IntIntervals (a,b) -> ( ( ) ( ) set ) equals :: TOPREALB:def 1
 { ].(a : ( ( ) ( ) set ) + n : ( ( ) ( complex real ext-real integer V66() ) Element of INT : ( ( ) ( non empty V27() V67() V68() V69() V70() V71() V73() ) set ) ) ) : ( ( ) ( ) set ) ,(b : ( ( non empty ) ( non empty ) set ) + n : ( ( ) ( complex real ext-real integer V66() ) Element of INT : ( ( ) ( non empty V27() V67() V68() V69() V70() V71() V73() ) set ) ) ) : ( ( ) ( ) set ) .[ : ( ( ) ( V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) where n is ( ( ) ( complex real ext-real integer V66() ) Element of INT : ( ( ) ( non empty V27() V67() V68() V69() V70() V71() V73() ) set ) ) : verum } ;
end;

theorem :: TOPREALB:3
for a, b being ( ( real ) ( complex real ext-real ) number )
for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in IntIntervals (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ) set ) iff ex n being ( ( ) ( complex real ext-real integer V66() ) Element of INT : ( ( ) ( non empty V27() V67() V68() V69() V70() V71() V73() ) set ) ) st x : ( ( ) ( ) set ) = ].(a : ( ( real ) ( complex real ext-real ) number ) + n : ( ( ) ( complex real ext-real integer V66() ) Element of INT : ( ( ) ( non empty V27() V67() V68() V69() V70() V71() V73() ) set ) ) ) : ( ( ) ( complex real ext-real ) set ) ,(b : ( ( real ) ( complex real ext-real ) number ) + n : ( ( ) ( complex real ext-real integer V66() ) Element of INT : ( ( ) ( non empty V27() V67() V68() V69() V70() V71() V73() ) set ) ) ) : ( ( ) ( complex real ext-real ) set ) .[ : ( ( ) ( open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

registration
let a, b be ( ( real ) ( complex real ext-real ) number ) ;
cluster IntIntervals (a : ( ( real ) ( complex real ext-real ) set ) ,b : ( ( real ) ( complex real ext-real ) set ) ) : ( ( ) ( ) set ) -> non empty ;
end;

theorem :: TOPREALB:4
for b, a being ( ( real ) ( complex real ext-real ) number ) st b : ( ( real ) ( complex real ext-real ) number ) - a : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( complex real ext-real ) set ) <= 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
IntIntervals (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( non empty ) set ) is mutually-disjoint ;

definition
let a, b be ( ( real ) ( complex real ext-real ) number ) ;
:: original: IntIntervals
redefine func IntIntervals (a,b) -> ( ( ) ( ) Subset-Family of ) ;
end;

definition
let a, b be ( ( real ) ( complex real ext-real ) number ) ;
:: original: IntIntervals
redefine func IntIntervals (a,b) -> ( ( open ) ( open ) Subset-Family of ) ;
end;

begin

definition
let r be ( ( real ) ( complex real ext-real ) number ) ;
func R^1 r -> ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) equals :: TOPREALB:def 2
r : ( ( ) ( ) set ) ;
end;

definition
let A be ( ( ) ( V67() V68() V69() ) Subset of ( ( ) ( non empty ) set ) ) ;
func R^1 A -> ( ( ) ( V67() V68() V69() ) Subset of ) equals :: TOPREALB:def 3
A : ( ( ) ( ) set ) ;
end;

registration
let A be ( ( non empty ) ( non empty V67() V68() V69() ) Subset of ( ( ) ( non empty ) set ) ) ;
cluster R^1 A : ( ( non empty ) ( non empty V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V67() V68() V69() ) Subset of ) -> non empty ;
end;

registration
let A be ( ( open ) ( open V67() V68() V69() ) Subset of ( ( ) ( non empty ) set ) ) ;
cluster R^1 A : ( ( open ) ( open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V67() V68() V69() ) Subset of ) -> open ;
end;

registration
let A be ( ( closed ) ( closed V67() V68() V69() ) Subset of ( ( ) ( non empty ) set ) ) ;
cluster R^1 A : ( ( closed ) ( closed V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V67() V68() V69() ) Subset of ) -> closed ;
end;

registration
let A be ( ( open ) ( open V67() V68() V69() ) Subset of ( ( ) ( non empty ) set ) ) ;
cluster R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 A : ( ( open ) ( open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() open ) Subset of ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) -> strict open ;
end;

registration
let A be ( ( closed ) ( closed V67() V68() V69() ) Subset of ( ( ) ( non empty ) set ) ) ;
cluster R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 A : ( ( closed ) ( closed V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() closed ) Subset of ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) -> strict closed ;
end;

definition
let f be ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like V51() V52() V53() ) PartFunc of ,) ;
func R^1 f -> ( ( Function-like quasi_total ) ( V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 (dom f : ( ( ) ( ) set ) ) : ( ( ) ( V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) V20( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 (rng f : ( ( ) ( ) set ) ) : ( ( ) ( V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) Function-like quasi_total V51() V52() V53() ) Function of ( ( ) ( V67() V68() V69() ) set ) , ( ( ) ( V67() V68() V69() ) set ) ) equals :: TOPREALB:def 4
f : ( ( ) ( ) set ) ;
end;

registration
let f be ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like V51() V52() V53() ) PartFunc of ,) ;
cluster R^1 f : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 (dom f : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) V20( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 (rng f : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) Function-like quasi_total V51() V52() V53() ) Function of ( ( ) ( V67() V68() V69() ) set ) , ( ( ) ( V67() V68() V69() ) set ) ) -> Function-like quasi_total onto ;
end;

registration
let f be ( ( Function-like one-to-one ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like one-to-one V51() V52() V53() ) PartFunc of ,) ;
cluster R^1 f : ( ( Function-like one-to-one ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like one-to-one V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 (dom f : ( ( Function-like one-to-one ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like one-to-one V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) V20( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 (rng f : ( ( Function-like one-to-one ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like one-to-one V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) Function-like quasi_total onto V51() V52() V53() ) Function of ( ( ) ( V67() V68() V69() ) set ) , ( ( ) ( V67() V68() V69() ) set ) ) -> Function-like one-to-one quasi_total ;
end;

theorem :: TOPREALB:5
R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ([#] REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty non proper V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() ) Subset of ) : ( ( strict ) ( non empty strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) = R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ;

theorem :: TOPREALB:6
for f being ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like V51() V52() V53() ) PartFunc of ,) st dom f : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like V51() V52() V53() ) PartFunc of ,) : ( ( ) ( V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) = REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) holds
R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 (dom f : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like V51() V52() V53() ) PartFunc of ,) ) : ( ( ) ( V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Subset of ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) = R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ;

theorem :: TOPREALB:7
for f being ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total V51() V52() V53() ) Function of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) , REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) holds f : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total V51() V52() V53() ) Function of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) , REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) is ( ( Function-like quasi_total ) ( V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 (rng b1 : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total V51() V52() V53() ) Function of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) , REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) Function-like quasi_total V51() V52() V53() ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( V67() V68() V69() ) set ) ) ;

theorem :: TOPREALB:8
for f being ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total V51() V52() V53() ) Function of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) , REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) holds f : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total V51() V52() V53() ) Function of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) , REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) is ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) Function-like total quasi_total V51() V52() V53() ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty V67() V68() V69() ) set ) ) ;

registration
let f be ( ( Function-like continuous ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like continuous V51() V52() V53() ) PartFunc of ,) ;
cluster R^1 f : ( ( Function-like continuous ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 (dom f : ( ( Function-like continuous ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) V20( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 (rng f : ( ( Function-like continuous ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) Function-like quasi_total onto V51() V52() V53() ) Function of ( ( ) ( V67() V68() V69() ) set ) , ( ( ) ( V67() V68() V69() ) set ) ) -> Function-like quasi_total continuous ;
end;

registration
let a be ( ( non zero real ) ( non zero complex real ext-real ) number ) ;
let b be ( ( real ) ( complex real ext-real ) number ) ;
cluster R^1 (AffineMap (a : ( ( non zero real ) ( non zero complex real ext-real ) set ) ,b : ( ( real ) ( complex real ext-real ) set ) )) : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 (dom (AffineMap (a : ( ( non zero real ) ( non zero complex real ext-real ) set ) ,b : ( ( real ) ( complex real ext-real ) set ) )) : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) V20( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 (rng (AffineMap (a : ( ( non zero real ) ( non zero complex real ext-real ) set ) ,b : ( ( real ) ( complex real ext-real ) set ) )) : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) Function-like one-to-one quasi_total onto bijective V51() V52() V53() continuous ) Function of ( ( ) ( V67() V68() V69() ) set ) , ( ( ) ( V67() V68() V69() ) set ) ) -> Function-like quasi_total open ;
end;

begin

definition
let S be ( ( ) ( TopSpace-like V221() V222() ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ;
attr S is being_simple_closed_curve means :: TOPREALB:def 5
 the carrier of S : ( ( ) ( ) set ) : ( ( ) ( ) set ) is ( ( being_simple_closed_curve ) ( non empty functional compact being_simple_closed_curve ) Simple_closed_curve) ;
end;

registration
cluster being_simple_closed_curve -> non empty compact pathwise_connected for ( ( ) ( ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ;
end;

registration
let r be ( ( real positive ) ( non empty complex real ext-real positive non negative ) number ) ;
let x be ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) ;
cluster Sphere (x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) ,r : ( ( real positive ) ( non empty complex real ext-real positive non negative ) set ) ) : ( ( ) ( non empty functional closed V258( TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) -> being_simple_closed_curve ;
end;

definition
let n be ( ( natural ) ( ordinal natural complex real ext-real non negative integer ) Nat) ;
let x be ( ( ) ( V16() Function-like V34(n : ( ( natural ) ( ordinal natural complex real ext-real non negative integer ) Nat) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) ;
let r be ( ( real ) ( complex real ext-real ) number ) ;
func Tcircle (x,r) -> ( ( ) ( TopSpace-like V221() V222() ) SubSpace of TOP-REAL n : ( ( ) ( ) set ) : ( ( V254() ) ( V254() ) L19()) ) equals :: TOPREALB:def 6
(TOP-REAL n : ( ( ) ( ) set ) ) : ( ( V254() ) ( V254() ) L19()) | (Sphere (x : ( ( non empty ) ( non empty ) set ) ,r : ( ( ) ( ) Element of n : ( ( ) ( ) set ) ) )) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL n : ( ( ) ( ) set ) ) : ( ( V254() ) ( V254() ) L19()) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( strict ) SubSpace of TOP-REAL n : ( ( ) ( ) set ) : ( ( V254() ) ( V254() ) L19()) ) ;
end;

registration
let n be ( ( non empty natural ) ( non empty ordinal natural complex real ext-real positive non negative integer ) Nat) ;
let x be ( ( ) ( V16() Function-like V34(n : ( ( non empty natural ) ( non empty ordinal natural complex real ext-real positive non negative integer ) Nat) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) ;
let r be ( ( real non negative ) ( complex real ext-real non negative ) number ) ;
cluster Tcircle (x : ( ( ) ( V16() Function-like V34(n : ( ( non empty natural ) ( non empty ordinal natural complex real ext-real positive non negative integer ) set ) ) V51() V52() V53() FinSequence-like ) Element of the carrier of (TOP-REAL n : ( ( non empty natural ) ( non empty ordinal natural complex real ext-real positive non negative integer ) set ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) ,r : ( ( real non negative ) ( complex real ext-real non negative ) set ) ) : ( ( ) ( TopSpace-like V221() V222() ) SubSpace of TOP-REAL n : ( ( non empty natural ) ( non empty ordinal natural complex real ext-real positive non negative integer ) set ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) -> non empty strict ;
end;

theorem :: TOPREALB:9
for n being ( ( ) ( ordinal natural complex real ext-real non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) )
for r being ( ( real ) ( complex real ext-real ) number )
for x being ( ( ) ( V16() Function-like V34(b1 : ( ( ) ( ordinal natural complex real ext-real non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) holds the carrier of (Tcircle (x : ( ( ) ( V16() Function-like V34(b1 : ( ( ) ( ordinal natural complex real ext-real non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) ,r : ( ( real ) ( complex real ext-real ) number ) )) : ( ( ) ( TopSpace-like V221() V222() ) SubSpace of TOP-REAL b1 : ( ( ) ( ordinal natural complex real ext-real non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( ) set ) = Sphere (x : ( ( ) ( V16() Function-like V34(b1 : ( ( ) ( ordinal natural complex real ext-real non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( functional closed V258( TOP-REAL b1 : ( ( ) ( ordinal natural complex real ext-real non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) Element of K6( the carrier of (TOP-REAL b1 : ( ( ) ( ordinal natural complex real ext-real non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ;

registration
let n be ( ( natural ) ( ordinal natural complex real ext-real non negative integer ) Nat) ;
let x be ( ( ) ( V16() Function-like V34(n : ( ( natural ) ( ordinal natural complex real ext-real non negative integer ) Nat) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) ;
let r be ( ( empty real ) ( empty trivial complex real ext-real non positive non negative Function-like functional V67() V68() V69() V70() V71() V72() V73() ) number ) ;
cluster Tcircle (x : ( ( ) ( V16() Function-like V34(n : ( ( natural ) ( ordinal natural complex real ext-real non negative integer ) set ) ) V51() V52() V53() FinSequence-like ) Element of the carrier of (TOP-REAL n : ( ( natural ) ( ordinal natural complex real ext-real non negative integer ) set ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) ,r : ( ( empty real ) ( empty trivial complex real ext-real non positive non negative Function-like functional V67() V68() V69() V70() V71() V72() V73() ) set ) ) : ( ( ) ( TopSpace-like V221() V222() ) SubSpace of TOP-REAL n : ( ( natural ) ( ordinal natural complex real ext-real non negative integer ) set ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) -> trivial ;
end;

theorem :: TOPREALB:10
for r being ( ( real ) ( complex real ext-real ) number ) holds Tcircle ((0. (TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like zero ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( TopSpace-like V221() V222() ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) is ( ( ) ( TopSpace-like V221() V222() ) SubSpace of Trectangle ((- r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( complex ) ( complex real ext-real ) set ) ,r : ( ( real ) ( complex real ext-real ) number ) ,(- r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( complex ) ( complex real ext-real ) set ) ,r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( TopSpace-like V221() V222() ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) ;

registration
let x be ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) ;
let r be ( ( real positive ) ( non empty complex real ext-real positive non negative ) number ) ;
cluster Tcircle (x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) ,r : ( ( real positive ) ( non empty complex real ext-real positive non negative ) set ) ) : ( ( ) ( non empty strict TopSpace-like V221() V222() ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) -> being_simple_closed_curve ;
end;

registration
cluster strict TopSpace-like V221() V222() being_simple_closed_curve for ( ( ) ( ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ;
end;

theorem :: TOPREALB:11
for S, T being ( ( being_simple_closed_curve ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) holds S : ( ( being_simple_closed_curve ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ,T : ( ( being_simple_closed_curve ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) are_homeomorphic ;

definition
let n be ( ( natural ) ( ordinal natural complex real ext-real non negative integer ) Nat) ;
func Tunit_circle n -> ( ( ) ( TopSpace-like V221() V222() ) SubSpace of TOP-REAL n : ( ( ) ( ) set ) : ( ( V254() ) ( V254() ) L19()) ) equals :: TOPREALB:def 7
 Tcircle ((0. (TOP-REAL n : ( ( ) ( ) set ) ) : ( ( V254() ) ( V254() ) L19()) ) : ( ( ) ( zero ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) set ) ) : ( ( V254() ) ( V254() ) L19()) : ( ( ) ( ) set ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( TopSpace-like V221() V222() ) SubSpace of TOP-REAL n : ( ( ) ( ) set ) : ( ( V254() ) ( V254() ) L19()) ) ;
end;

registration
let n be ( ( non empty natural ) ( non empty ordinal natural complex real ext-real positive non negative integer ) Nat) ;
cluster Tunit_circle n : ( ( non empty natural ) ( non empty ordinal natural complex real ext-real positive non negative integer ) set ) : ( ( ) ( TopSpace-like V221() V222() ) SubSpace of TOP-REAL n : ( ( non empty natural ) ( non empty ordinal natural complex real ext-real positive non negative integer ) set ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) -> non empty ;
end;

theorem :: TOPREALB:12
for n being ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) )
for x being ( ( ) ( V16() Function-like V34(b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) st x : ( ( ) ( V16() Function-like V34(b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) is ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) holds
|.x : ( ( ) ( V16() Function-like V34(b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) .| : ( ( ) ( complex real ext-real non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) = 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: TOPREALB:13
for x being ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) st x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) is ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) holds
( - 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty complex real ext-real non positive negative integer ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) <= x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) & x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) <= 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) & - 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty complex real ext-real non positive negative integer ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) <= x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) & x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) <= 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TOPREALB:14
for x being ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) st x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) is ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) & x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) = 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) = 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: TOPREALB:15
for x being ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) st x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) is ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) & x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) = - 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty complex real ext-real non positive negative integer ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) holds
x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) = 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: TOPREALB:16
for x being ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) st x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) is ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) & x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) = 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) = 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: TOPREALB:17
for x being ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) st x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) is ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) & x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) = - 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty complex real ext-real non positive negative integer ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) holds
x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) = 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: TOPREALB:18
Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like V221() V222() ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) is ( ( ) ( TopSpace-like V221() V222() ) SubSpace of Trectangle ((- 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real non positive negative integer ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,(- 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real non positive negative integer ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like V221() V222() ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) ;

theorem :: TOPREALB:19
for n being ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) )
for r being ( ( real positive ) ( non empty complex real ext-real positive non negative ) number )
for x being ( ( ) ( V16() Function-like V34(b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (Tunit_circle b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like V221() V222() ) SubSpace of TOP-REAL b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of (Tcircle (b3 : ( ( ) ( V16() Function-like V34(b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) ,b2 : ( ( real positive ) ( non empty complex real ext-real positive non negative ) number ) )) : ( ( ) ( non empty strict TopSpace-like V221() V222() ) SubSpace of TOP-REAL b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st ( for a being ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) )
for b being ( ( ) ( V16() Function-like V34(b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) st a : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) = b : ( ( ) ( V16() Function-like V34(b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) holds
f : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (Tunit_circle b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like V221() V222() ) SubSpace of TOP-REAL b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of (Tcircle (b3 : ( ( ) ( V16() Function-like V34(b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) ,b2 : ( ( real positive ) ( non empty complex real ext-real positive non negative ) number ) )) : ( ( ) ( non empty strict TopSpace-like V221() V222() ) SubSpace of TOP-REAL b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . a : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tcircle (b3 : ( ( ) ( V16() Function-like V34(b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) ,b2 : ( ( real positive ) ( non empty complex real ext-real positive non negative ) number ) )) : ( ( ) ( non empty strict TopSpace-like V221() V222() ) SubSpace of TOP-REAL b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) = (r : ( ( real positive ) ( non empty complex real ext-real positive non negative ) number ) * b : ( ( ) ( V16() Function-like V34(b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( V16() Function-like V34(b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Element of the carrier of (TOP-REAL b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) + x : ( ( ) ( V16() Function-like V34(b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V16() Function-like V34(b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Element of the carrier of (TOP-REAL b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) ) holds
f : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (Tunit_circle b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like V221() V222() ) SubSpace of TOP-REAL b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of (Tcircle (b3 : ( ( ) ( V16() Function-like V34(b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) ,b2 : ( ( real positive ) ( non empty complex real ext-real positive non negative ) number ) )) : ( ( ) ( non empty strict TopSpace-like V221() V222() ) SubSpace of TOP-REAL b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is being_homeomorphism ;

registration
cluster Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like V221() V222() ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) -> being_simple_closed_curve ;
end;

theorem :: TOPREALB:20
for n being ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) )
for r, s being ( ( real positive ) ( non empty complex real ext-real positive non negative ) number )
for x, y being ( ( ) ( V16() Function-like V34(b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) holds Tcircle (x : ( ( ) ( V16() Function-like V34(b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) ,r : ( ( real positive ) ( non empty complex real ext-real positive non negative ) number ) ) : ( ( ) ( non empty strict TopSpace-like V221() V222() ) SubSpace of TOP-REAL b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) , Tcircle (y : ( ( ) ( V16() Function-like V34(b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) ,s : ( ( real positive ) ( non empty complex real ext-real positive non negative ) number ) ) : ( ( ) ( non empty strict TopSpace-like V221() V222() ) SubSpace of TOP-REAL b1 : ( ( non empty ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) are_homeomorphic ;

registration
let x be ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) ;
let r be ( ( real non negative ) ( complex real ext-real non negative ) number ) ;
cluster Tcircle (x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) ,r : ( ( real non negative ) ( complex real ext-real non negative ) set ) ) : ( ( ) ( non empty strict TopSpace-like V221() V222() ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) -> pathwise_connected ;
end;

definition
func c[10] -> ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) equals :: TOPREALB:def 8
|[1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ]| : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) ;
func c[-10] -> ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) equals :: TOPREALB:def 9
|[(- 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real non positive negative integer ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ]| : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) ;
end;

definition
let p be ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ;
func Topen_unit_circle p -> ( ( strict ) ( strict TopSpace-like V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) means :: TOPREALB:def 10
 the carrier of it : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) = the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) \ {p : ( ( ) ( ) set ) } : ( ( ) ( non empty trivial ) Element of K6( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ;
end;

registration
let p be ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ;
cluster Topen_unit_circle p : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( strict TopSpace-like V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) -> non empty strict ;
end;

theorem :: TOPREALB:21
for p being ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) holds p : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) is not ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TOPREALB:22
for p being ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) holds Topen_unit_circle p : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict TopSpace-like V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) = (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) | (([#] (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty non proper open closed dense non boundary ) Element of K6( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) \ {p : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty trivial functional compact ) Element of K6( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( strict TopSpace-like V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) ;

theorem :: TOPREALB:23
for p, q being ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) <> q : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) holds
q : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) is ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TOPREALB:24
for p being ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) st p : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) is ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) = 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
p : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) = c[-10] : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TOPREALB:25
for p being ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) st p : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) is ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) = 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
p : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) = c[10] : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TOPREALB:26
for p being ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) )
for x being ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) st x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) is ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) holds
( - 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty complex real ext-real non positive negative integer ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) <= x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) & x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) <= 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) & - 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty complex real ext-real non positive negative integer ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) <= x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) & x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) <= 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TOPREALB:27
for x being ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) st x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) is ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) holds
( - 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty complex real ext-real non positive negative integer ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) <= x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) & x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) < 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TOPREALB:28
for x being ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) st x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) is ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) holds
( - 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty complex real ext-real non positive negative integer ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) < x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) & x : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) <= 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ;

registration
let p be ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ;
cluster Topen_unit_circle p : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict TopSpace-like V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) -> strict open ;
end;

theorem :: TOPREALB:29
for p being ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) holds Topen_unit_circle p : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) , I(01) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of K744() : ( ( ) ( non empty strict TopSpace-like V299() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) are_homeomorphic ;

theorem :: TOPREALB:30
for p, q being ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) holds Topen_unit_circle p : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) , Topen_unit_circle q : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) are_homeomorphic ;

begin

definition
func CircleMap -> ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) means :: TOPREALB:def 11
 for x being ( ( real ) ( complex real ext-real ) number ) holds it : ( ( ) ( ) set ) . x : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( ) set ) = |[(cos ((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) * x : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,(sin ((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) * x : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ]| : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) ;
end;

theorem :: TOPREALB:31
for i being ( ( integer ) ( complex real ext-real integer ) Integer)
for r being ( ( real ) ( complex real ext-real ) number ) holds CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . r : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( ) set ) = CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . (r : ( ( real ) ( complex real ext-real ) number ) + i : ( ( integer ) ( complex real ext-real integer ) Integer) ) : ( ( ) ( complex real ext-real ) set ) : ( ( ) ( ) set ) ;

theorem :: TOPREALB:32
for i being ( ( integer ) ( complex real ext-real integer ) Integer) holds CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . i : ( ( integer ) ( complex real ext-real integer ) Integer) : ( ( ) ( ) set ) = c[10] : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TOPREALB:33
CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) " {c[10] : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty trivial functional compact ) Element of K6( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V67() V68() V69() ) Element of K6( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) : ( ( ) ( non empty ) set ) ) = INT : ( ( ) ( non empty V27() V67() V68() V69() V70() V71() V73() ) set ) ;

theorem :: TOPREALB:34
for r being ( ( real ) ( complex real ext-real ) number ) st frac r : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) = 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) holds
CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . r : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( ) set ) = |[(- 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real non positive negative integer ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ]| : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) ;

theorem :: TOPREALB:35
for r being ( ( real ) ( complex real ext-real ) number ) st frac r : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) = 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) holds
CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . r : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( ) set ) = |[0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ]| : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) ;

theorem :: TOPREALB:36
for r being ( ( real ) ( complex real ext-real ) number ) st frac r : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) = 3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 4 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) holds
CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . r : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( ) set ) = |[0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,(- 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real non positive negative integer ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ]| : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) ;

theorem :: TOPREALB:37
for r being ( ( real ) ( complex real ext-real ) number )
for i, j being ( ( integer ) ( complex real ext-real integer ) Integer) holds CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . r : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( ) set ) = |[((cos : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) * (AffineMap ((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) * i : ( ( integer ) ( complex real ext-real integer ) Integer) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) . r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) ,((sin : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) * (AffineMap ((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,((2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) * j : ( ( integer ) ( complex real ext-real integer ) Integer) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like total quasi_total continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) . r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) set ) ]| : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) ;

registration
cluster CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) -> Function-like quasi_total continuous ;
end;

theorem :: TOPREALB:38
for A being ( ( ) ( V67() V68() V69() ) Subset of )
for f being ( ( Function-like quasi_total ) ( V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | b1 : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) st [.0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) .[ : ( ( ) ( non empty V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) c= A : ( ( ) ( V67() V68() V69() ) Subset of ) & f : ( ( Function-like quasi_total ) ( V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | b1 : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) = CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) | A : ( ( ) ( V67() V68() V69() ) Subset of ) : ( ( Function-like ) ( V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like ) Element of K6(K7( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) , the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like quasi_total ) ( V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | b1 : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) is onto ;

registration
cluster CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) -> Function-like quasi_total onto ;
end;

registration
let r be ( ( real ) ( complex real ext-real ) number ) ;
cluster K69(CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) ,[.r : ( ( real ) ( complex real ext-real ) set ) ,(r : ( ( real ) ( complex real ext-real ) set ) + 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V16() ) ( V16() Function-like ) set ) -> V16() one-to-one ;
end;

registration
let r be ( ( real ) ( complex real ext-real ) number ) ;
cluster K69(CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) ,].r : ( ( real ) ( complex real ext-real ) set ) ,(r : ( ( real ) ( complex real ext-real ) set ) + 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V16() ) ( V16() Function-like ) set ) -> V16() one-to-one ;
end;

theorem :: TOPREALB:39
for b, a being ( ( real ) ( complex real ext-real ) number ) st b : ( ( real ) ( complex real ext-real ) number ) - a : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( complex real ext-real ) set ) <= 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
for d being ( ( ) ( ) set ) st d : ( ( ) ( ) set ) in IntIntervals (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ) : ( ( open ) ( non empty open ) Subset-Family of ) holds
CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) | d : ( ( ) ( ) set ) : ( ( Function-like ) ( V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like ) Element of K6(K7( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) , the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) is one-to-one ;

theorem :: TOPREALB:40
for a, b being ( ( real ) ( complex real ext-real ) number )
for d being ( ( ) ( ) set ) st d : ( ( ) ( ) set ) in IntIntervals (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) ) : ( ( open ) ( non empty open ) Subset-Family of ) holds
CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) .: d : ( ( ) ( ) set ) : ( ( ) ( ) Element of K6( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) = CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) .: (union (IntIntervals (a : ( ( real ) ( complex real ext-real ) number ) ,b : ( ( real ) ( complex real ext-real ) number ) )) : ( ( open ) ( non empty open ) Subset-Family of ) ) : ( ( ) ( V67() V68() V69() ) Element of K6( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ;

definition
let r be ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ;
func CircleMap r -> ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].r : ( ( ) ( ) set ) ,(r : ( ( ) ( ) set ) + 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) V20( the carrier of (Topen_unit_circle (CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . r : ( ( ) ( ) set ) ) : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) equals :: TOPREALB:def 12
CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) | ].r : ( ( ) ( ) set ) ,(r : ( ( ) ( ) set ) + 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like ) Element of K6(K7( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) , the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: TOPREALB:41
for i being ( ( integer ) ( complex real ext-real integer ) Integer)
for a being ( ( real ) ( complex real ext-real ) number ) holds CircleMap (R^1 (a : ( ( real ) ( complex real ext-real ) number ) + i : ( ( integer ) ( complex real ext-real integer ) Integer) ) : ( ( ) ( complex real ext-real ) set ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].(R^1 (b2 : ( ( real ) ( complex real ext-real ) number ) + b1 : ( ( integer ) ( complex real ext-real integer ) Integer) ) : ( ( ) ( complex real ext-real ) set ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ,((R^1 (b2 : ( ( real ) ( complex real ext-real ) number ) + b1 : ( ( integer ) ( complex real ext-real integer ) Integer) ) : ( ( ) ( complex real ext-real ) set ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) + 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Topen_unit_circle (CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . (R^1 (b2 : ( ( real ) ( complex real ext-real ) number ) + b1 : ( ( integer ) ( complex real ext-real integer ) Integer) ) : ( ( ) ( complex real ext-real ) set ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ) : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) = (CircleMap (R^1 a : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].(R^1 b2 : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ,((R^1 b2 : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) + 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Topen_unit_circle (CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . (R^1 b2 : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ) : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) * ((AffineMap (1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,(- i : ( ( integer ) ( complex real ext-real integer ) Integer) ) : ( ( complex ) ( complex real ext-real integer ) set ) )) : ( ( Function-like quasi_total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like one-to-one total quasi_total onto bijective continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) | ].(a : ( ( real ) ( complex real ext-real ) number ) + i : ( ( integer ) ( complex real ext-real integer ) Integer) ) : ( ( ) ( complex real ext-real ) set ) ,((a : ( ( real ) ( complex real ext-real ) number ) + i : ( ( integer ) ( complex real ext-real integer ) Integer) ) : ( ( ) ( complex real ext-real ) set ) + 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) Function-like continuous V51() V52() V53() ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ,REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V16() V19( REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) V20( the carrier of (Topen_unit_circle (CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . (R^1 b2 : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ) : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) , the carrier of (Topen_unit_circle (CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . (R^1 b2 : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ) : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ;

registration
let r be ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ;
cluster CircleMap r : ( ( ) ( complex real ext-real ) Element of the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].r : ( ( ) ( complex real ext-real ) Element of the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) ,(r : ( ( ) ( complex real ext-real ) Element of the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) + 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Topen_unit_circle (CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . r : ( ( ) ( complex real ext-real ) Element of the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) ) : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) -> Function-like one-to-one quasi_total onto continuous ;
end;

definition
func Circle2IntervalR -> ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (Topen_unit_circle c[10] : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) Function-like total quasi_total V51() V52() V53() ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V67() V68() V69() ) set ) ) means :: TOPREALB:def 13
 for p being ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ex x, y being ( ( real ) ( complex real ext-real ) number ) st
( p : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) = |[x : ( ( real ) ( complex real ext-real ) number ) ,y : ( ( real ) ( complex real ext-real ) number ) ]| : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) & ( y : ( ( real ) ( complex real ext-real ) number ) >= 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) implies it : ( ( ) ( ) set ) . p : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) Element of the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) = (arccos x : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) & ( y : ( ( real ) ( complex real ext-real ) number ) <= 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) implies it : ( ( ) ( ) set ) . p : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) Element of the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) = 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) - ((arccos x : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) );
end;

definition
func Circle2IntervalL -> ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (Topen_unit_circle c[-10] : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].(1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,(3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) Function-like total quasi_total V51() V52() V53() ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V67() V68() V69() ) set ) ) means :: TOPREALB:def 14
 for p being ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ex x, y being ( ( real ) ( complex real ext-real ) number ) st
( p : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) = |[x : ( ( real ) ( complex real ext-real ) number ) ,y : ( ( real ) ( complex real ext-real ) number ) ]| : ( ( ) ( V16() Function-like V34(2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V51() V52() V53() FinSequence-like ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) : ( ( ) ( non empty functional ) set ) ) & ( y : ( ( real ) ( complex real ext-real ) number ) >= 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) implies it : ( ( ) ( ) set ) . p : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) Element of the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].(1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,(3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) = 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) + ((arccos x : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) & ( y : ( ( real ) ( complex real ext-real ) number ) <= 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) implies it : ( ( ) ( ) set ) . p : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) Element of the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].(1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,(3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) = 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) - ((arccos x : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) / (2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) * PI : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) );
end;

theorem :: TOPREALB:42
(CircleMap (R^1 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].(R^1 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ,((R^1 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) + 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Topen_unit_circle (CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . (R^1 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ) : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) Function-like one-to-one total quasi_total onto bijective continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) " : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (Topen_unit_circle (CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . (R^1 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ) : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].(R^1 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ,((R^1 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) + 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) Function-like total quasi_total V51() V52() V53() ) Element of K6(K7( the carrier of (Topen_unit_circle (CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . (R^1 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ) : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) , the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].(R^1 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ,((R^1 0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) + 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) = Circle2IntervalR : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (Topen_unit_circle c[10] : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) Function-like total quasi_total V51() V52() V53() ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V67() V68() V69() ) set ) ) ;

theorem :: TOPREALB:43
(CircleMap (R^1 (1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].(R^1 (1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ,((R^1 (1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) + 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Topen_unit_circle (CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . (R^1 (1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ) : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) Function-like one-to-one total quasi_total onto bijective continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) " : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (Topen_unit_circle (CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . (R^1 (1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ) : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].(R^1 (1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ,((R^1 (1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) + 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) Function-like total quasi_total V51() V52() V53() ) Element of K6(K7( the carrier of (Topen_unit_circle (CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . (R^1 (1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ) : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) , the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].(R^1 (1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ,((R^1 (1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) + 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) : ( ( ) ( non empty V51() V52() V53() ) set ) ) : ( ( ) ( non empty ) set ) ) = Circle2IntervalL : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (Topen_unit_circle c[-10] : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].(1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,(3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) Function-like total quasi_total V51() V52() V53() ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V67() V68() V69() ) set ) ) ;

registration
cluster Circle2IntervalR : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (Topen_unit_circle c[10] : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) Function-like total quasi_total V51() V52() V53() ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V67() V68() V69() ) set ) ) -> Function-like one-to-one quasi_total onto continuous ;
cluster Circle2IntervalL : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (Topen_unit_circle c[-10] : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].(1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,(3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) Function-like total quasi_total V51() V52() V53() ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V67() V68() V69() ) set ) ) -> Function-like one-to-one quasi_total onto continuous ;
end;

registration
let i be ( ( integer ) ( complex real ext-real integer ) Integer) ;
cluster CircleMap (R^1 i : ( ( integer ) ( complex real ext-real integer ) set ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].(R^1 i : ( ( integer ) ( complex real ext-real integer ) set ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ,((R^1 i : ( ( integer ) ( complex real ext-real integer ) set ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) + 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Topen_unit_circle (CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . (R^1 i : ( ( integer ) ( complex real ext-real integer ) set ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ) : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) Function-like one-to-one total quasi_total onto bijective continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) -> Function-like quasi_total open ;
cluster CircleMap (R^1 ((1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) + i : ( ( integer ) ( complex real ext-real integer ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].(R^1 ((1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) + i : ( ( integer ) ( complex real ext-real integer ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ,((R^1 ((1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) + i : ( ( integer ) ( complex real ext-real integer ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) + 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Topen_unit_circle (CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . (R^1 ((1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) + i : ( ( integer ) ( complex real ext-real integer ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ) : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) Function-like one-to-one total quasi_total onto bijective continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) -> Function-like quasi_total open ;
end;

registration
cluster Circle2IntervalR : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (Topen_unit_circle c[10] : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) Function-like one-to-one total quasi_total onto bijective V51() V52() V53() continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V67() V68() V69() ) set ) ) -> Function-like quasi_total open ;
cluster Circle2IntervalL : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (Topen_unit_circle c[-10] : ( ( ) ( V16() Function-like FinSequence-like ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) V20( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].(1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,(3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) Function-like one-to-one total quasi_total onto bijective V51() V52() V53() continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V67() V68() V69() ) set ) ) -> Function-like quasi_total open ;
end;

theorem :: TOPREALB:44
CircleMap (R^1 (1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | (R^1 ].(R^1 (1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ,((R^1 (1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) + 1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V67() V68() V69() open ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like open V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Topen_unit_circle (CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) . (R^1 (1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( ) ( complex real ext-real ) Point of ( ( ) ( non empty V67() V68() V69() ) set ) ) ) : ( ( ) ( V16() Function-like FinSequence-like ) Element of the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict TopSpace-like open V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( non empty ) set ) ) Function-like one-to-one total quasi_total onto bijective continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) is being_homeomorphism ;

theorem :: TOPREALB:45
ex F being ( ( ) ( ) Subset-Family of ) st
( F : ( ( ) ( ) Subset-Family of ) = {(CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) .: ].0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ,(CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) .: ].(1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,(3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) Element of K6(K6( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & F : ( ( ) ( ) Subset-Family of ) is ( ( ) ( ) Cover of ( ( ) ( non empty ) set ) ) & F : ( ( ) ( ) Subset-Family of ) is open & ( for U being ( ( ) ( ) Subset of ) holds
( ( U : ( ( ) ( ) Subset of ) = CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) .: ].0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) implies ( union (IntIntervals (0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( open ) ( non empty open ) Subset-Family of ) : ( ( ) ( V67() V68() V69() ) Element of K6( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) : ( ( ) ( non empty ) set ) ) = CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) " U : ( ( ) ( ) Subset of ) : ( ( ) ( V67() V68() V69() ) Element of K6( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) : ( ( ) ( non empty ) set ) ) & ( for d being ( ( ) ( V67() V68() V69() ) Subset of ) st d : ( ( ) ( V67() V68() V69() ) Subset of ) in IntIntervals (0 : ( ( ) ( empty trivial ordinal natural complex real ext-real non positive non negative Function-like functional integer V66() V67() V68() V69() V70() V71() V72() V73() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( open ) ( non empty open ) Subset-Family of ) holds
for f being ( ( Function-like quasi_total ) ( V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | b3 : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) V20( the carrier of ((Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) | b2 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Function of ( ( ) ( V67() V68() V69() ) set ) , ( ( ) ( ) set ) ) st f : ( ( Function-like quasi_total ) ( V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | b3 : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) V20( the carrier of ((Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) | b2 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Function of ( ( ) ( V67() V68() V69() ) set ) , ( ( ) ( ) set ) ) = CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) | d : ( ( ) ( V67() V68() V69() ) Subset of ) : ( ( Function-like ) ( V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like ) Element of K6(K7( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) , the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like quasi_total ) ( V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | b3 : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) V20( the carrier of ((Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) | b2 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Function of ( ( ) ( V67() V68() V69() ) set ) , ( ( ) ( ) set ) ) is being_homeomorphism ) ) ) & ( U : ( ( ) ( ) Subset of ) = CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) .: ].(1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,(3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) .[ : ( ( ) ( non empty open V67() V68() V69() ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) implies ( union (IntIntervals ((1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,(3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) )) : ( ( open ) ( non empty open ) Subset-Family of ) : ( ( ) ( V67() V68() V69() ) Element of K6( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) : ( ( ) ( non empty ) set ) ) = CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) " U : ( ( ) ( ) Subset of ) : ( ( ) ( V67() V68() V69() ) Element of K6( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) : ( ( ) ( non empty ) set ) ) & ( for d being ( ( ) ( V67() V68() V69() ) Subset of ) st d : ( ( ) ( V67() V68() V69() ) Subset of ) in IntIntervals ((1 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ,(3 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) ) : ( ( open ) ( non empty open ) Subset-Family of ) holds
for f being ( ( Function-like quasi_total ) ( V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | b3 : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) V20( the carrier of ((Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) | b2 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Function of ( ( ) ( V67() V68() V69() ) set ) , ( ( ) ( ) set ) ) st f : ( ( Function-like quasi_total ) ( V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | b3 : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) V20( the carrier of ((Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) | b2 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Function of ( ( ) ( V67() V68() V69() ) set ) , ( ( ) ( ) set ) ) = CircleMap : ( ( Function-like quasi_total ) ( non empty V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like total quasi_total onto continuous ) Function of ( ( ) ( non empty V67() V68() V69() ) set ) , ( ( ) ( non empty ) set ) ) | d : ( ( ) ( V67() V68() V69() ) Subset of ) : ( ( Function-like ) ( V16() V19( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) ) V20( the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) Function-like ) Element of K6(K7( the carrier of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) : ( ( ) ( non empty V67() V68() V69() ) set ) , the carrier of (Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like quasi_total ) ( V16() V19( the carrier of (R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) | b3 : ( ( ) ( V67() V68() V69() ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V299() ) SubSpace of R^1 : ( ( V303() ) ( non empty strict TopSpace-like V299() V303() ) SubSpace of R^1 : ( ( TopSpace-like ) ( non empty strict TopSpace-like V299() ) TopStruct ) ) ) : ( ( ) ( V67() V68() V69() ) set ) ) V20( the carrier of ((Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) | b2 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like V221() V222() ) SubSpace of Tunit_circle 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty TopSpace-like compact V221() V222() V266() pathwise_connected being_simple_closed_curve ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty ordinal natural complex real ext-real positive non negative integer V66() V67() V68() V69() V70() V71() V72() ) Element of NAT : ( ( ) ( V67() V68() V69() V70() V71() V72() V73() non with_non-empty_elements ) Element of K6(REAL : ( ( ) ( non empty V27() V67() V68() V69() V73() non with_non-empty_elements ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( V254() ) ( non empty TopSpace-like T_0 T_1 T_2 V136() V161() V162() V163() V164() V165() V166() V167() V221() V222() V254() ) L19()) ) ) : ( ( ) ( ) set ) ) Function-like quasi_total ) Function of ( ( ) ( V67() V68() V69() ) set ) , ( ( ) ( ) set ) ) is being_homeomorphism ) ) ) ) ) ) ;