:: SIN_COS6 semantic presentation

begin

theorem :: SIN_COS6:1
for r, s being ( ( real ) ( V22() real ext-real ) number ) st 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 ) ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < s : ( ( real ) ( V22() real ext-real ) number ) holds
[\(r : ( ( real ) ( V22() real ext-real ) number ) / s : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) set ) /] : ( ( integer ) ( V22() real ext-real integer ) 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 ) ) ) ;

theorem :: SIN_COS6:2
for f being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function)
for X, Y being ( ( ) ( ) set ) st f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) | X : ( ( ) ( ) set ) : ( ( Relation-like ) ( Relation-like Function-like ) set ) is one-to-one & Y : ( ( ) ( ) set ) c= X : ( ( ) ( ) set ) holds
f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) | Y : ( ( ) ( ) set ) : ( ( Relation-like ) ( Relation-like Function-like ) set ) is one-to-one ;

begin

theorem :: SIN_COS6:3
for r being ( ( real ) ( V22() real ext-real ) number ) holds - 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 ) ) <= sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) set ) ;

theorem :: SIN_COS6:4
for r being ( ( real ) ( V22() real ext-real ) number ) holds sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

theorem :: SIN_COS6:5
for r being ( ( real ) ( V22() real ext-real ) number ) holds - 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 ) ) <= cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) set ) ;

theorem :: SIN_COS6:6
for r being ( ( real ) ( V22() real ext-real ) number ) holds cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

registration
cluster PI : ( ( real ) ( V22() real ext-real ) set ) -> real positive ;
end;

theorem :: SIN_COS6:7
( sin (- (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 ) ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) & 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 ) ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) : ( ( ) ( non empty V22() real ext-real non positive negative integer ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) ;

theorem :: SIN_COS6:8
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) holds 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 ) ) . r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) = 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 ) ) . (r : ( ( real ) ( V22() real ext-real ) number ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ;

theorem :: SIN_COS6:9
( cos (- (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 ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) 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 ) ) ) & 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 ) ) . (- (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 ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) 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 ) ) ) ) ;

theorem :: SIN_COS6:10
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) holds 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 ) ) . r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) = 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 ) ) . (r : ( ( real ) ( V22() real ext-real ) number ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ;

theorem :: SIN_COS6:11
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) < r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

theorem :: SIN_COS6:12
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) < r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

theorem :: SIN_COS6:13
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st (- (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 ) ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) < r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < (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 ) ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

theorem :: SIN_COS6:14
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st (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 ) ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) < r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < ((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

theorem :: SIN_COS6:15
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st ((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) < r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

theorem :: SIN_COS6:16
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

theorem :: SIN_COS6:17
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

theorem :: SIN_COS6:18
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st (- (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 ) ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= (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 ) ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

theorem :: SIN_COS6:19
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st (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 ) ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= ((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

theorem :: SIN_COS6:20
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st ((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

theorem :: SIN_COS6:21
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) & sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) & not r : ( ( real ) ( V22() real ext-real ) number ) = (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
r : ( ( real ) ( V22() real ext-real ) number ) = 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ;

theorem :: SIN_COS6:22
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) & cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) & not r : ( ( real ) ( V22() real ext-real ) number ) = (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 ) ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
r : ( ( real ) ( V22() real ext-real ) number ) = ((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ;

theorem :: SIN_COS6:23
for r being ( ( real ) ( V22() real ext-real ) number ) st sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) holds
r : ( ( real ) ( V22() real ext-real ) number ) = ((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * [\(r : ( ( real ) ( V22() real ext-real ) number ) / (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) /] : ( ( integer ) ( V22() real ext-real integer ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ;

theorem :: SIN_COS6:24
for r being ( ( real ) ( V22() real ext-real ) number ) st sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) holds
r : ( ( real ) ( V22() real ext-real ) number ) = (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 ) ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * [\(r : ( ( real ) ( V22() real ext-real ) number ) / (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) /] : ( ( integer ) ( V22() real ext-real integer ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ;

theorem :: SIN_COS6:25
for r being ( ( real ) ( V22() real ext-real ) number ) st cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) holds
r : ( ( real ) ( V22() real ext-real ) number ) = 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * [\(r : ( ( real ) ( V22() real ext-real ) number ) / (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) /] : ( ( integer ) ( V22() real ext-real integer ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ;

theorem :: SIN_COS6:26
for r being ( ( real ) ( V22() real ext-real ) number ) st cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) holds
r : ( ( real ) ( V22() real ext-real ) number ) = (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * [\(r : ( ( real ) ( V22() real ext-real ) number ) / (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) /] : ( ( integer ) ( V22() real ext-real integer ) set ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ;

theorem :: SIN_COS6:27
for r being ( ( real ) ( V22() real ext-real ) number ) st 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 ) ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) & sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) holds
r : ( ( real ) ( V22() real ext-real ) number ) = (3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ;

theorem :: SIN_COS6:28
for r being ( ( real ) ( V22() real ext-real ) number ) st 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 ) ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) & sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) holds
r : ( ( real ) ( V22() real ext-real ) number ) = 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 ) ) ;

theorem :: SIN_COS6:29
for r being ( ( real ) ( V22() real ext-real ) number ) st 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 ) ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) & cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) holds
r : ( ( real ) ( V22() real ext-real ) number ) = PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ;

theorem :: SIN_COS6:30
for r being ( ( real ) ( V22() real ext-real ) number ) st 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 ) ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < 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 ) ) holds
sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

theorem :: SIN_COS6:31
for r being ( ( real ) ( V22() real ext-real ) number ) st 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 ) ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < (3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ;

theorem :: SIN_COS6:32
for r being ( ( real ) ( V22() real ext-real ) number ) st (3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) < r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ;

theorem :: SIN_COS6:33
for r being ( ( real ) ( V22() real ext-real ) number ) st 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 ) ) < r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

theorem :: SIN_COS6:34
for r being ( ( real ) ( V22() real ext-real ) number ) st 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 ) ) ) < r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

theorem :: SIN_COS6:35
for r being ( ( real ) ( V22() real ext-real ) number ) st 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 ) ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ;

theorem :: SIN_COS6:36
for r being ( ( real ) ( V22() real ext-real ) number ) st PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) < r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ;

theorem :: SIN_COS6:37
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < (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 ) ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

theorem :: SIN_COS6:38
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < ((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ;

theorem :: SIN_COS6:39
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st ((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) < r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ;

theorem :: SIN_COS6:40
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st (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 ) ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) < r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

theorem :: SIN_COS6:41
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) < r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) ;

theorem :: SIN_COS6:42
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ;

theorem :: SIN_COS6:43
for r being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) < r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) 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 ) ) ;

theorem :: SIN_COS6:44
for r being ( ( real ) ( V22() real ext-real ) number ) st cos ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) holds
r : ( ( real ) ( V22() real ext-real ) number ) in INT : ( ( ) ( V70() V71() V72() V73() V74() V76() ) set ) ;

theorem :: SIN_COS6:45
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 ) ) : ( ( ) ( 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 ) ) ;

theorem :: SIN_COS6:46
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 ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( 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 ) ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS6:47
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 ) ) ,((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) 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 ) ) : ( ( ) ( 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 ) ) ;

theorem :: SIN_COS6:48
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 ) ) ,((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( 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 ) ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS6:49
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 ) ) : ( ( ) ( 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 ) ) ;

theorem :: SIN_COS6:50
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 ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( 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 ) ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS6:51
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 ) ) .: [.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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) 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 ) ) : ( ( ) ( 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 ) ) ;

theorem :: SIN_COS6:52
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 ) ) .: ].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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( 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 ) ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS6:53
for i being ( ( integer ) ( V22() real ext-real integer ) number ) holds 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 ) ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) 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 ) ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) 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 V46() V47() V48() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) is increasing ;

theorem :: SIN_COS6:54
for i being ( ( integer ) ( V22() real ext-real integer ) number ) holds 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 ) ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,(((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) 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 V46() V47() V48() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) is decreasing ;

theorem :: SIN_COS6:55
for i being ( ( integer ) ( V22() real ext-real integer ) number ) holds 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 ) ) | [.((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) 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 V46() V47() V48() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) is decreasing ;

theorem :: SIN_COS6:56
for i being ( ( integer ) ( V22() real ext-real integer ) number ) holds 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 ) ) | [.(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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) 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 V46() V47() V48() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) is increasing ;

theorem :: SIN_COS6:57
for i being ( ( integer ) ( V22() real ext-real integer ) number ) holds 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 ) ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) 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 ) ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) 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 V46() V47() V48() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) is one-to-one ;

theorem :: SIN_COS6:58
for i being ( ( integer ) ( V22() real ext-real integer ) number ) holds 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 ) ) + ((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,(((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) 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 V46() V47() V48() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) is one-to-one ;

registration
cluster 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 ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) ,((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) 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 ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
end;

registration
cluster 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 ) ) ,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 ) ) ) .] : ( ( ) ( V28() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) | [.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 ) ) / 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 ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) ,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 ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) ,((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) 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 ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) | [.((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) 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 ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
end;

registration
cluster 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 ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) ,((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) ,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 ) ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) | ].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 ) ) / 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 ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) ,PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) ,((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) | ].((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
end;

theorem :: SIN_COS6:59
for i being ( ( integer ) ( V22() real ext-real integer ) number ) holds 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 ) ) | [.((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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) 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 V46() V47() V48() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) is one-to-one ;

theorem :: SIN_COS6:60
for i being ( ( integer ) ( V22() real ext-real integer ) number ) holds 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 ) ) | [.(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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) 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 V46() V47() V48() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) is one-to-one ;

registration
cluster 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 ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) | [.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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) 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 ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
end;

registration
cluster 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 ) ) / 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 ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) | [.(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 ) ) ,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 ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) | [.PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) 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 ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) | [.((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) 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 ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
end;

registration
cluster 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 ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) | ].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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) / 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 ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) | ].(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 ) ) ,PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) | ].PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ,((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
cluster 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 ) ) | ].((3 : ( ( ) ( 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 ) ) ) / 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 ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) .[ : ( ( ) ( V29() V70() V71() V72() ) Element of K19(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like ) ( Relation-like Function-like V46() V47() V48() ) set ) -> Relation-like one-to-one ;
end;

theorem :: SIN_COS6:61
for r, s being ( ( real ) ( V22() real ext-real ) number )
for i being ( ( integer ) ( V22() real ext-real integer ) number ) st (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) <= s : ( ( real ) ( V22() real ext-real ) number ) & s : ( ( real ) ( V22() real ext-real ) number ) < (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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( 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 ) ) ) * PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) * i : ( ( integer ) ( V22() real ext-real integer ) number ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) & sin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) set ) = sin s : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) set ) & cos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) set ) = cos s : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) set ) holds
r : ( ( real ) ( V22() real ext-real ) number ) = s : ( ( real ) ( V22() real ext-real ) number ) ;

begin

definition
func arcsin -> ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -defined REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -valued Function-like V46() V47() V48() ) PartFunc of ,) equals :: SIN_COS6:def 1
(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 ) ) " : ( ( 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() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
let r be ( ( ) ( ) set ) ;
func arcsin r -> ( ( ) ( ) set ) equals :: SIN_COS6:def 2
arcsin : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -defined REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -valued Function-like V46() V47() V48() ) PartFunc of ,) . r : ( ( ) ( ) set ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ;
end;

definition
let r be ( ( ) ( ) set ) ;
:: original: arcsin
redefine func arcsin r -> ( ( ) ( V22() real ext-real ) Real) ;
end;

theorem :: SIN_COS6:62
rng arcsin : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -defined REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -valued Function-like V46() V47() V48() ) PartFunc of ,) : ( ( ) ( 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 ) ) ;

registration
cluster arcsin : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -defined REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -valued Function-like V46() V47() V48() ) PartFunc of ,) -> Function-like one-to-one ;
end;

theorem :: SIN_COS6:63
dom 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 ,) : ( ( ) ( 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 ) ) ;

theorem :: SIN_COS6:64
(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 :: SIN_COS6:65
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_COS6:66
(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 :: SIN_COS6:67
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 ) ) ;

theorem :: SIN_COS6:68
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) holds
sin (arcsin r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) Real) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) = r : ( ( real ) ( V22() real ext-real ) number ) ;

theorem :: SIN_COS6:69
for r being ( ( real ) ( V22() real ext-real ) number ) st - (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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) holds
arcsin (sin r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) set ) : ( ( ) ( V22() real ext-real ) Real) = r : ( ( real ) ( V22() real ext-real ) number ) ;

theorem :: SIN_COS6:70
arcsin (- 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 ) ) : ( ( ) ( V22() real ext-real ) Real) = - (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 ) ) ;

theorem :: SIN_COS6:71
arcsin 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 ) ) ) : ( ( ) ( V22() real ext-real ) Real) = 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 ) ) ) ;

theorem :: SIN_COS6:72
arcsin 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 ) ) ) : ( ( ) ( V22() real ext-real ) Real) = 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 ) ) ;

theorem :: SIN_COS6:73
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) & arcsin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Real) = - (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 ) ) holds
r : ( ( real ) ( V22() real ext-real ) number ) = - 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 ) ) ;

theorem :: SIN_COS6:74
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) & arcsin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Real) = 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 ) ) ) holds
r : ( ( real ) ( V22() real ext-real ) number ) = 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 ) ) ) ;

theorem :: SIN_COS6:75
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) & arcsin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Real) = 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 ) ) holds
r : ( ( real ) ( V22() real ext-real ) number ) = 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 ) ) ) ;

theorem :: SIN_COS6:76
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) holds
( - (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 ) ) <= arcsin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Real) & arcsin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Real) <= 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 ) ) ) ;

theorem :: SIN_COS6:77
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) < r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < 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 ) ) ) holds
( - (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 ) ) < arcsin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Real) & arcsin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Real) < 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 ) ) ) ;

theorem :: SIN_COS6:78
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) holds
arcsin r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Real) = - (arcsin (- r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( V22() ) ( V22() real ext-real ) set ) ) : ( ( ) ( V22() real ext-real ) Real) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ;

theorem :: SIN_COS6:79
for s, r being ( ( real ) ( V22() real ext-real ) number ) st 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 ) ) ) <= s : ( ( real ) ( V22() real ext-real ) number ) & (r : ( ( real ) ( V22() real ext-real ) number ) ^2) : ( ( ) ( V22() real ext-real ) set ) + (s : ( ( real ) ( V22() real ext-real ) number ) ^2) : ( ( ) ( V22() real ext-real ) set ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) holds
cos (arcsin r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) Real) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) = s : ( ( real ) ( V22() real ext-real ) number ) ;

theorem :: SIN_COS6:80
for s, r being ( ( real ) ( V22() real ext-real ) number ) st s : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) & (r : ( ( real ) ( V22() real ext-real ) number ) ^2) : ( ( ) ( V22() real ext-real ) set ) + (s : ( ( real ) ( V22() real ext-real ) number ) ^2) : ( ( ) ( V22() real ext-real ) set ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) holds
cos (arcsin r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) Real) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) = - s : ( ( real ) ( V22() real ext-real ) number ) : ( ( V22() ) ( V22() real ext-real ) set ) ;

theorem :: SIN_COS6:81
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) holds
cos (arcsin r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) Real) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) = sqrt (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 ) ) ) - (r : ( ( real ) ( V22() real ext-real ) number ) ^2) : ( ( ) ( V22() real ext-real ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ;

theorem :: SIN_COS6:82
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 ,) | [.(- 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 ) ) : ( ( ) ( 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 ) ) is increasing ;

theorem :: SIN_COS6:83
for r being ( ( real ) ( V22() real ext-real ) number ) holds
( 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 ,) is_differentiable_on ].(- 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 ) ) ) .[ : ( ( ) ( V29() 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 ) ) < r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < 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 ) ) ) implies diff (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 ,) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) / (sqrt (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 ) ) ) - (r : ( ( real ) ( V22() real ext-real ) number ) ^2) : ( ( ) ( V22() real ext-real ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) ) ;

theorem :: SIN_COS6:84
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 ,) | [.(- 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 ) ) : ( ( ) ( 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 ) ) is continuous ;

begin

definition
func arccos -> ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -defined REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -valued Function-like V46() V47() V48() ) PartFunc of ,) equals :: SIN_COS6:def 3
(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 ) ) " : ( ( 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() ) Element of K19(K20(REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ,REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V46() V47() V48() ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
let r be ( ( ) ( ) set ) ;
func arccos r -> ( ( ) ( ) set ) equals :: SIN_COS6:def 4
arccos : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -defined REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -valued Function-like V46() V47() V48() ) PartFunc of ,) . r : ( ( ) ( ) set ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ;
end;

definition
let r be ( ( ) ( ) set ) ;
:: original: arccos
redefine func arccos r -> ( ( ) ( V22() real ext-real ) Real) ;
end;

theorem :: SIN_COS6:85
rng arccos : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -defined REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -valued Function-like V46() V47() V48() ) PartFunc of ,) : ( ( ) ( 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 ) ) ;

registration
cluster arccos : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -defined REAL : ( ( ) ( V70() V71() V72() V76() ) set ) -valued Function-like V46() V47() V48() ) PartFunc of ,) -> Function-like one-to-one ;
end;

theorem :: SIN_COS6:86
dom 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 ,) : ( ( ) ( 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 ) ) ;

theorem :: SIN_COS6:87
(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 :: SIN_COS6:88
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 :: SIN_COS6:89
(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 :: SIN_COS6:90
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 ) ) ;

theorem :: SIN_COS6:91
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) holds
cos (arccos r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) Real) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) = r : ( ( real ) ( V22() real ext-real ) number ) ;

theorem :: SIN_COS6:92
for r being ( ( real ) ( V22() real ext-real ) number ) st 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 ) ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
arccos (cos r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) set ) : ( ( ) ( V22() real ext-real ) Real) = r : ( ( real ) ( V22() real ext-real ) number ) ;

theorem :: SIN_COS6:93
arccos (- 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 ) ) : ( ( ) ( V22() real ext-real ) Real) = PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ;

theorem :: SIN_COS6:94
arccos 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 ) ) ) : ( ( ) ( V22() real ext-real ) Real) = 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 ) ) ;

theorem :: SIN_COS6:95
arccos 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 ) ) ) : ( ( ) ( V22() real ext-real ) Real) = 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 ) ) ) ;

theorem :: SIN_COS6:96
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) & arccos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Real) = 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 ) ) ) holds
r : ( ( real ) ( V22() real ext-real ) number ) = 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 ) ) ) ;

theorem :: SIN_COS6:97
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) & arccos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Real) = 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 ) ) holds
r : ( ( real ) ( V22() real ext-real ) number ) = 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 ) ) ) ;

theorem :: SIN_COS6:98
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) & arccos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Real) = PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) holds
r : ( ( real ) ( V22() real ext-real ) number ) = - 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 ) ) ;

theorem :: SIN_COS6:99
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) holds
( 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 ) ) ) <= arccos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Real) & arccos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Real) <= PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) ;

theorem :: SIN_COS6:100
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) < r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < 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 ) ) ) holds
( 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 ) ) ) < arccos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Real) & arccos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Real) < PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) ;

theorem :: SIN_COS6:101
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) holds
arccos r : ( ( real ) ( V22() real ext-real ) number ) : ( ( ) ( V22() real ext-real ) Real) = PI : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) - (arccos (- r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( V22() ) ( V22() real ext-real ) set ) ) : ( ( ) ( V22() real ext-real ) Real) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ;

theorem :: SIN_COS6:102
for s, r being ( ( real ) ( V22() real ext-real ) number ) st 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 ) ) ) <= s : ( ( real ) ( V22() real ext-real ) number ) & (r : ( ( real ) ( V22() real ext-real ) number ) ^2) : ( ( ) ( V22() real ext-real ) set ) + (s : ( ( real ) ( V22() real ext-real ) number ) ^2) : ( ( ) ( V22() real ext-real ) set ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) holds
sin (arccos r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) Real) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) = s : ( ( real ) ( V22() real ext-real ) number ) ;

theorem :: SIN_COS6:103
for s, r being ( ( real ) ( V22() real ext-real ) number ) st s : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) & (r : ( ( real ) ( V22() real ext-real ) number ) ^2) : ( ( ) ( V22() real ext-real ) set ) + (s : ( ( real ) ( V22() real ext-real ) number ) ^2) : ( ( ) ( V22() real ext-real ) set ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) holds
sin (arccos r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) Real) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) = - s : ( ( real ) ( V22() real ext-real ) number ) : ( ( V22() ) ( V22() real ext-real ) set ) ;

theorem :: SIN_COS6:104
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) holds
sin (arccos r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) Real) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) = sqrt (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 ) ) ) - (r : ( ( real ) ( V22() real ext-real ) number ) ^2) : ( ( ) ( V22() real ext-real ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ;

theorem :: SIN_COS6:105
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 ,) | [.(- 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 ) ) : ( ( ) ( 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 ) ) is decreasing ;

theorem :: SIN_COS6:106
for r being ( ( real ) ( V22() real ext-real ) number ) holds
( 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 ,) is_differentiable_on ].(- 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 ) ) ) .[ : ( ( ) ( V29() 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 ) ) < r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) < 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 ) ) ) implies diff (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 ,) ,r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) 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 ) ) ) / (sqrt (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 ) ) ) - (r : ( ( real ) ( V22() real ext-real ) number ) ^2) : ( ( ) ( V22() real ext-real ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( V70() V71() V72() V76() ) set ) ) ) ) ;

theorem :: SIN_COS6:107
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 ,) | [.(- 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 ) ) : ( ( ) ( 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 ) ) is continuous ;

theorem :: SIN_COS6:108
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) holds
(arcsin r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) Real) + (arccos r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) Real) : ( ( ) ( V22() real ext-real ) 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 ) ) ;

theorem :: SIN_COS6:109
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) holds
(arccos (- r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( V22() ) ( V22() real ext-real ) set ) ) : ( ( ) ( V22() real ext-real ) Real) - (arcsin r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) Real) : ( ( ) ( V22() real ext-real ) 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 ) ) ;

theorem :: SIN_COS6:110
for r being ( ( real ) ( V22() real ext-real ) number ) st - 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 ) ) <= r : ( ( real ) ( V22() real ext-real ) number ) & r : ( ( real ) ( V22() real ext-real ) number ) <= 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 ) ) ) holds
(arccos r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) Real) - (arcsin (- r : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( V22() ) ( V22() real ext-real ) set ) ) : ( ( ) ( V22() real ext-real ) Real) : ( ( ) ( V22() real ext-real ) 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 ) ) ;