begin
begin
begin
definition
let r be ( ( ) ( )
set ) ;
end;
theorem
(sin : ( ( Function-like V33( REAL : ( ( ) ( V70() V71() V72() V76() ) set ) , REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) ( Relation-like REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -defined REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -valued Function-like V33( REAL : ( ( ) ( V70() V71() V72() V76() ) set ) , REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) continuous V46() V47() V48() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) | [.(- (PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,(PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) ( V28() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
Element of
K19(
K20(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ,
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) )
* arcsin : ( (
Function-like ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
PartFunc of ,) : ( (
Relation-like ) (
Relation-like Function-like one-to-one V46()
V47()
V48() )
set )
= id [.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
total ) (
Relation-like [.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) )
-defined [.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like one-to-one total V46()
V47()
V48() )
Element of
K19(
K20(
[.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) ,
[.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
arcsin : ( (
Function-like ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
PartFunc of ,)
* (sin : ( ( Function-like V33( REAL : ( ( ) ( V70() V71() V72() V76() ) set ) , REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) ( Relation-like REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -defined REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -valued Function-like V33( REAL : ( ( ) ( V70() V71() V72() V76() ) set ) , REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) continuous V46() V47() V48() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) | [.(- (PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,(PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) ( V28() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
Element of
K19(
K20(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ,
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
Element of
K19(
K20(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ,
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) )
= id [.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
total ) (
Relation-like [.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) )
-defined [.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like one-to-one total V46()
V47()
V48() )
Element of
K19(
K20(
[.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) ,
[.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
(sin : ( ( Function-like V33( REAL : ( ( ) ( V70() V71() V72() V76() ) set ) , REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) ( Relation-like REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -defined REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -valued Function-like V33( REAL : ( ( ) ( V70() V71() V72() V76() ) set ) , REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) continuous V46() V47() V48() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) | [.(- (PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,(PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) ( V28() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
Element of
K19(
K20(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ,
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) )
* arcsin : ( (
Function-like ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
PartFunc of ,) : ( ( ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
Element of
K19(
K20(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ,
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) )
= id [.(- (PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,(PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
total ) (
Relation-like [.(- (PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,(PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) )
-defined [.(- (PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,(PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like one-to-one total V46()
V47()
V48() )
Element of
K19(
K20(
[.(- (PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,(PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) ,
[.(- (PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,(PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
arcsin : ( (
Function-like ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
PartFunc of ,)
* (sin : ( ( Function-like V33( REAL : ( ( ) ( V70() V71() V72() V76() ) set ) , REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) ( Relation-like REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -defined REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -valued Function-like V33( REAL : ( ( ) ( V70() V71() V72() V76() ) set ) , REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) continuous V46() V47() V48() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) | [.(- (PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,(PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) ( V28() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
Element of
K19(
K20(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ,
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Relation-like ) (
Relation-like Function-like one-to-one V46()
V47()
V48() )
set )
= id [.(- (PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,(PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
total ) (
Relation-like [.(- (PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,(PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) )
-defined [.(- (PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,(PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like one-to-one total V46()
V47()
V48() )
Element of
K19(
K20(
[.(- (PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,(PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) ,
[.(- (PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,(PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) ) ;
begin
definition
let r be ( ( ) ( )
set ) ;
end;
theorem
(cos : ( ( Function-like V33( REAL : ( ( ) ( V70() V71() V72() V76() ) set ) , REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) ( Relation-like REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -defined REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -valued Function-like V33( REAL : ( ( ) ( V70() V71() V72() V76() ) set ) , REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) continuous V46() V47() V48() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) | [.0 : ( ( ) ( Function-like functional empty natural V22() real ext-real non positive non negative integer V69() V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) ( V28() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
Element of
K19(
K20(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ,
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) )
* arccos : ( (
Function-like ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
PartFunc of ,) : ( (
Relation-like ) (
Relation-like Function-like one-to-one V46()
V47()
V48() )
set )
= id [.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
total ) (
Relation-like [.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) )
-defined [.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like one-to-one total V46()
V47()
V48() )
Element of
K19(
K20(
[.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) ,
[.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
arccos : ( (
Function-like ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
PartFunc of ,)
* (cos : ( ( Function-like V33( REAL : ( ( ) ( V70() V71() V72() V76() ) set ) , REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) ( Relation-like REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -defined REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -valued Function-like V33( REAL : ( ( ) ( V70() V71() V72() V76() ) set ) , REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) continuous V46() V47() V48() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) | [.0 : ( ( ) ( Function-like functional empty natural V22() real ext-real non positive non negative integer V69() V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) ( V28() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
Element of
K19(
K20(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ,
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
Element of
K19(
K20(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ,
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) )
= id [.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
total ) (
Relation-like [.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) )
-defined [.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like one-to-one total V46()
V47()
V48() )
Element of
K19(
K20(
[.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) ,
[.(- 1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,1 : ( ( ) ( non empty natural V22() real ext-real positive non negative integer V69() V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
(cos : ( ( Function-like V33( REAL : ( ( ) ( V70() V71() V72() V76() ) set ) , REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) ( Relation-like REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -defined REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -valued Function-like V33( REAL : ( ( ) ( V70() V71() V72() V76() ) set ) , REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) continuous V46() V47() V48() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) | [.0 : ( ( ) ( Function-like functional empty natural V22() real ext-real non positive non negative integer V69() V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) ( V28() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
Element of
K19(
K20(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ,
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) )
* arccos : ( (
Function-like ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
PartFunc of ,) : ( ( ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
Element of
K19(
K20(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ,
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) )
= id [.0 : ( ( ) ( Function-like functional empty natural V22() real ext-real non positive non negative integer V69() V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
total ) (
Relation-like [.0 : ( ( ) ( Function-like functional empty natural V22() real ext-real non positive non negative integer V69() V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) )
-defined [.0 : ( ( ) ( Function-like functional empty natural V22() real ext-real non positive non negative integer V69() V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like one-to-one total V46()
V47()
V48() )
Element of
K19(
K20(
[.0 : ( ( ) ( Function-like functional empty natural V22() real ext-real non positive non negative integer V69() V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) ,
[.0 : ( ( ) ( Function-like functional empty natural V22() real ext-real non positive non negative integer V69() V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
arccos : ( (
Function-like ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
PartFunc of ,)
* (cos : ( ( Function-like V33( REAL : ( ( ) ( V70() V71() V72() V76() ) set ) , REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) ( Relation-like REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -defined REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -valued Function-like V33( REAL : ( ( ) ( V70() V71() V72() V76() ) set ) , REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) continuous V46() V47() V48() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) | [.0 : ( ( ) ( Function-like functional empty natural V22() real ext-real non positive non negative integer V69() V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) ( V28() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
Relation-like REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-defined REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set )
-valued Function-like one-to-one V46()
V47()
V48() )
Element of
K19(
K20(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ,
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Relation-like ) (
Relation-like Function-like one-to-one V46()
V47()
V48() )
set )
= id [.0 : ( ( ) ( Function-like functional empty natural V22() real ext-real non positive non negative integer V69() V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
total ) (
Relation-like [.0 : ( ( ) ( Function-like functional empty natural V22() real ext-real non positive non negative integer V69() V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) )
-defined [.0 : ( ( ) ( Function-like functional empty natural V22() real ext-real non positive non negative integer V69() V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like one-to-one total V46()
V47()
V48() )
Element of
K19(
K20(
[.0 : ( ( ) ( Function-like functional empty natural V22() real ext-real non positive non negative integer V69() V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) ,
[.0 : ( ( ) ( Function-like functional empty natural V22() real ext-real non positive non negative integer V69() V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( V70() V71() V72() V73() V74() V75() V76() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .] : ( ( ) (
V28()
V70()
V71()
V72() )
Element of
K19(
REAL : ( ( ) (
V70()
V71()
V72()
V76() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V46()
V47()
V48() )
set ) ) : ( ( ) ( )
set ) ) ;