:: DIFF_2 semantic presentation

begin

theorem :: DIFF_2:1
for x, h being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) holds [!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x : ( ( ) ( V11() V12() real ) Real) ,(x : ( ( ) ( V11() V12() real ) Real) + h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = (((fdif (f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,K81(REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined K81(REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) : ( ( ) ( ) set ) -valued Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,K81(REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) : ( ( ) ( ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,K81(REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) . 1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) set ) / h : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:2
for h, x being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st h : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) holds
[!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x : ( ( ) ( V11() V12() real ) Real) ,(x : ( ( ) ( V11() V12() real ) Real) + h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ,(x : ( ( ) ( V11() V12() real ) Real) + (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = (((fdif (f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,K81(REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined K81(REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) : ( ( ) ( ) set ) -valued Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,K81(REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) : ( ( ) ( ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,K81(REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) . 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) set ) / (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * (h : ( ( ) ( V11() V12() real ) Real) ^2) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:3
for x, h being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) holds [!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,(x : ( ( ) ( V11() V12() real ) Real) - h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ,x : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = (((bdif (f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,K81(REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined K81(REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) : ( ( ) ( ) set ) -valued Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,K81(REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) : ( ( ) ( ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,K81(REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) . 1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) set ) / h : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:4
for h, x being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st h : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) holds
[!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,(x : ( ( ) ( V11() V12() real ) Real) - (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ,(x : ( ( ) ( V11() V12() real ) Real) - h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ,x : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = (((bdif (f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,K81(REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined K81(REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) : ( ( ) ( ) set ) -valued Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,K81(REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) : ( ( ) ( ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,K81(REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) . 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) set ) / (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * (h : ( ( ) ( V11() V12() real ) Real) ^2) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:5
for r, x0, x1, x2 being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) holds [!(r : ( ( ) ( V11() V12() real ) Real) (#) f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = r : ( ( ) ( V11() V12() real ) Real) * [!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:6
for x0, x1, x2 being ( ( ) ( V11() V12() real ) Real)
for f1, f2 being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) holds [!(f1 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) + f2 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = [!f1 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + [!f2 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:7
for r1, r2, x0, x1, x2 being ( ( ) ( V11() V12() real ) Real)
for f1, f2 being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) holds [!((r1 : ( ( ) ( V11() V12() real ) Real) (#) f1 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) + (r2 : ( ( ) ( V11() V12() real ) Real) (#) f2 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = (r1 : ( ( ) ( V11() V12() real ) Real) * [!f1 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + (r2 : ( ( ) ( V11() V12() real ) Real) * [!f2 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:8
for r, x0, x1, x2, x3 being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) holds [!(r : ( ( ) ( V11() V12() real ) Real) (#) f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = r : ( ( ) ( V11() V12() real ) Real) * [!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:9
for x0, x1, x2, x3 being ( ( ) ( V11() V12() real ) Real)
for f1, f2 being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) holds [!(f1 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) + f2 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = [!f1 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + [!f2 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:10
for r1, r2, x0, x1, x2, x3 being ( ( ) ( V11() V12() real ) Real)
for f1, f2 being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) holds [!((r1 : ( ( ) ( V11() V12() real ) Real) (#) f1 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) + (r2 : ( ( ) ( V11() V12() real ) Real) (#) f2 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = (r1 : ( ( ) ( V11() V12() real ) Real) * [!f1 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + (r2 : ( ( ) ( V11() V12() real ) Real) * [!f2 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

definition
let f be ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) Function) ;
let x0, x1, x2, x3, x4 be ( ( real ) ( V11() V12() real ) number ) ;
func [!f,x0,x1,x2,x3,x4!] -> ( ( ) ( V11() V12() real ) Real) equals :: DIFF_2:def 1
([!f : ( ( real ) ( V11() V12() real ) set ) ,x0 : ( ( real ) ( V11() V12() real ) set ) ,x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,x2 : ( ( real ) ( V11() V12() real ) set ) ,x3 : ( ( real ) ( V11() V12() real ) set ) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - [!f : ( ( real ) ( V11() V12() real ) set ) ,x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,x2 : ( ( real ) ( V11() V12() real ) set ) ,x3 : ( ( real ) ( V11() V12() real ) set ) ,x4 : ( ( Function-like V30(x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,x2 : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) -defined x2 : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26(x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V30(x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,x2 : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,x2 : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / (x0 : ( ( real ) ( V11() V12() real ) set ) - x4 : ( ( Function-like V30(x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,x2 : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) -defined x2 : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26(x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V30(x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,x2 : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,x2 : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ) set ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;
end;

theorem :: DIFF_2:11
for r, x0, x1, x2, x3, x4 being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) holds [!(r : ( ( ) ( V11() V12() real ) Real) (#) f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) = r : ( ( ) ( V11() V12() real ) Real) * [!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:12
for x0, x1, x2, x3, x4 being ( ( ) ( V11() V12() real ) Real)
for f1, f2 being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) holds [!(f1 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) + f2 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) = [!f1 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) + [!f2 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:13
for r1, r2, x0, x1, x2, x3, x4 being ( ( ) ( V11() V12() real ) Real)
for f1, f2 being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) holds [!((r1 : ( ( ) ( V11() V12() real ) Real) (#) f1 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) + (r2 : ( ( ) ( V11() V12() real ) Real) (#) f2 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) = (r1 : ( ( ) ( V11() V12() real ) Real) * [!f1 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + (r2 : ( ( ) ( V11() V12() real ) Real) * [!f2 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

definition
let f be ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) Function) ;
let x0, x1, x2, x3, x4, x5 be ( ( real ) ( V11() V12() real ) number ) ;
func [!f,x0,x1,x2,x3,x4,x5!] -> ( ( ) ( V11() V12() real ) Real) equals :: DIFF_2:def 2
([!f : ( ( real ) ( V11() V12() real ) set ) ,x0 : ( ( real ) ( V11() V12() real ) set ) ,x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,x2 : ( ( real ) ( V11() V12() real ) set ) ,x3 : ( ( real ) ( V11() V12() real ) set ) ,x4 : ( ( Function-like V30(x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,x2 : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) -defined x2 : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26(x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V30(x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,x2 : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,x2 : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) !] : ( ( ) ( V11() V12() real ) Real) - [!f : ( ( real ) ( V11() V12() real ) set ) ,x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,x2 : ( ( real ) ( V11() V12() real ) set ) ,x3 : ( ( real ) ( V11() V12() real ) set ) ,x4 : ( ( Function-like V30(x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,x2 : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) -defined x2 : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26(x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V30(x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,x2 : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:x1 : ( ( Function-like V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) ( Relation-like NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) -defined f : ( ( real ) ( V11() V12() real ) set ) -valued Function-like V26( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ) V30( NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) ) ) M2( bool [:NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) ,f : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,x2 : ( ( real ) ( V11() V12() real ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,x5 : ( ( ) ( ) set ) !] : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / (x0 : ( ( real ) ( V11() V12() real ) set ) - x5 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;
end;

theorem :: DIFF_2:14
for r, x0, x1, x2, x3, x4, x5 being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) holds [!(r : ( ( ) ( V11() V12() real ) Real) (#) f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) ,x5 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) = r : ( ( ) ( V11() V12() real ) Real) * [!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) ,x5 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:15
for x0, x1, x2, x3, x4, x5 being ( ( ) ( V11() V12() real ) Real)
for f1, f2 being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) holds [!(f1 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) + f2 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) ,x5 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) = [!f1 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) ,x5 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) + [!f2 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) ,x5 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:16
for r1, r2, x0, x1, x2, x3, x4, x5 being ( ( ) ( V11() V12() real ) Real)
for f1, f2 being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) holds [!((r1 : ( ( ) ( V11() V12() real ) Real) (#) f1 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) + (r2 : ( ( ) ( V11() V12() real ) Real) (#) f2 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) ,x5 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) = (r1 : ( ( ) ( V11() V12() real ) Real) * [!f1 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) ,x5 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + (r2 : ( ( ) ( V11() V12() real ) Real) * [!f2 : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) ,x5 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:17
for x0, x1, x2 being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) are_mutually_different holds
[!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = (((f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) . x0 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / ((x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (x0 : ( ( ) ( V11() V12() real ) Real) - x2 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + ((f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) . x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / ((x1 : ( ( ) ( V11() V12() real ) Real) - x0 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (x1 : ( ( ) ( V11() V12() real ) Real) - x2 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + ((f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) . x2 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / ((x2 : ( ( ) ( V11() V12() real ) Real) - x0 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (x2 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:18
for x0, x1, x2, x3 being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) are_mutually_different holds
( [!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = [!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x0 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) & [!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = [!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x0 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) ;

theorem :: DIFF_2:19
for x0, x1, x2, x3 being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) are_mutually_different holds
( [!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = [!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) & [!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = [!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) ;

theorem :: DIFF_2:20
for x0, x1, x2 being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) is constant holds
[!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ;

theorem :: DIFF_2:21
for x0, x1, a, b being ( ( ) ( V11() V12() real ) Real) st x0 : ( ( ) ( V11() V12() real ) Real) <> x1 : ( ( ) ( V11() V12() real ) Real) holds
[!(AffineMap (a : ( ( ) ( V11() V12() real ) Real) ,b : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = a : ( ( ) ( V11() V12() real ) Real) ;

theorem :: DIFF_2:22
for x0, x1, x2, a, b being ( ( ) ( V11() V12() real ) Real) st x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) are_mutually_different holds
[!(AffineMap (a : ( ( ) ( V11() V12() real ) Real) ,b : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ;

theorem :: DIFF_2:23
for x0, x1, x2, x3, a, b being ( ( ) ( V11() V12() real ) Real) st x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) are_mutually_different holds
[!(AffineMap (a : ( ( ) ( V11() V12() real ) Real) ,b : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ;

theorem :: DIFF_2:24
for a, b, h, x being ( ( ) ( V11() V12() real ) Real) holds (fD ((AffineMap (a : ( ( ) ( V11() V12() real ) Real) ,b : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = a : ( ( ) ( V11() V12() real ) Real) * h : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:25
for a, b, h, x being ( ( ) ( V11() V12() real ) Real) holds (bD ((AffineMap (a : ( ( ) ( V11() V12() real ) Real) ,b : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = a : ( ( ) ( V11() V12() real ) Real) * h : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:26
for a, b, h, x being ( ( ) ( V11() V12() real ) Real) holds (cD ((AffineMap (a : ( ( ) ( V11() V12() real ) Real) ,b : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = a : ( ( ) ( V11() V12() real ) Real) * h : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:27
for a, b, c, x0, x1 being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st ( for x being ( ( ) ( V11() V12() real ) Real) holds f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = ((a : ( ( ) ( V11() V12() real ) Real) * (x : ( ( ) ( V11() V12() real ) Real) ^2) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + (b : ( ( ) ( V11() V12() real ) Real) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + c : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) & x0 : ( ( ) ( V11() V12() real ) Real) <> x1 : ( ( ) ( V11() V12() real ) Real) holds
[!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = (a : ( ( ) ( V11() V12() real ) Real) * (x0 : ( ( ) ( V11() V12() real ) Real) + x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + b : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:28
for a, b, c, x0, x1, x2 being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st ( for x being ( ( ) ( V11() V12() real ) Real) holds f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = ((a : ( ( ) ( V11() V12() real ) Real) * (x : ( ( ) ( V11() V12() real ) Real) ^2) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + (b : ( ( ) ( V11() V12() real ) Real) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + c : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) & x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) are_mutually_different holds
[!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = a : ( ( ) ( V11() V12() real ) Real) ;

theorem :: DIFF_2:29
for a, b, c, x0, x1, x2, x3 being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st ( for x being ( ( ) ( V11() V12() real ) Real) holds f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = ((a : ( ( ) ( V11() V12() real ) Real) * (x : ( ( ) ( V11() V12() real ) Real) ^2) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + (b : ( ( ) ( V11() V12() real ) Real) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + c : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) & x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) are_mutually_different holds
[!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ;

theorem :: DIFF_2:30
for a, b, c, x0, x1, x2, x3, x4 being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st ( for x being ( ( ) ( V11() V12() real ) Real) holds f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = ((a : ( ( ) ( V11() V12() real ) Real) * (x : ( ( ) ( V11() V12() real ) Real) ^2) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + (b : ( ( ) ( V11() V12() real ) Real) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + c : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) & x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) are_mutually_different holds
[!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) = 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ;

theorem :: DIFF_2:31
for a, b, c, h being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st ( for x being ( ( ) ( V11() V12() real ) Real) holds f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = ((a : ( ( ) ( V11() V12() real ) Real) * (x : ( ( ) ( V11() V12() real ) Real) ^2) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + (b : ( ( ) ( V11() V12() real ) Real) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + c : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) holds
for x being ( ( ) ( V11() V12() real ) Real) holds (fD (f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = ((((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * a : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + (a : ( ( ) ( V11() V12() real ) Real) * (h : ( ( ) ( V11() V12() real ) Real) ^2) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + (b : ( ( ) ( V11() V12() real ) Real) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:32
for a, b, c, h being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st ( for x being ( ( ) ( V11() V12() real ) Real) holds f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = ((a : ( ( ) ( V11() V12() real ) Real) * (x : ( ( ) ( V11() V12() real ) Real) ^2) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + (b : ( ( ) ( V11() V12() real ) Real) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + c : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) holds
for x being ( ( ) ( V11() V12() real ) Real) holds (bD (f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = ((((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * a : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - (a : ( ( ) ( V11() V12() real ) Real) * (h : ( ( ) ( V11() V12() real ) Real) ^2) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + (b : ( ( ) ( V11() V12() real ) Real) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:33
for a, b, c, h being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st ( for x being ( ( ) ( V11() V12() real ) Real) holds f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = ((a : ( ( ) ( V11() V12() real ) Real) * (x : ( ( ) ( V11() V12() real ) Real) ^2) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + (b : ( ( ) ( V11() V12() real ) Real) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + c : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) holds
for x being ( ( ) ( V11() V12() real ) Real) holds (cD (f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * a : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + (b : ( ( ) ( V11() V12() real ) Real) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:34
for k, x0, x1 being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st ( for x being ( ( ) ( V11() V12() real ) Real) holds f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = k : ( ( ) ( V11() V12() real ) Real) / x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) & x0 : ( ( ) ( V11() V12() real ) Real) <> x1 : ( ( ) ( V11() V12() real ) Real) & x0 : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) & x1 : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) holds
[!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = - (k : ( ( ) ( V11() V12() real ) Real) / (x0 : ( ( ) ( V11() V12() real ) Real) * x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:35
for k, x0, x1, x2 being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st ( for x being ( ( ) ( V11() V12() real ) Real) holds f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = k : ( ( ) ( V11() V12() real ) Real) / x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) & x0 : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) & x1 : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) & x2 : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) & x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) are_mutually_different holds
[!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = k : ( ( ) ( V11() V12() real ) Real) / ((x0 : ( ( ) ( V11() V12() real ) Real) * x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * x2 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:36
for k, x0, x1, x2, x3 being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st ( for x being ( ( ) ( V11() V12() real ) Real) holds f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = k : ( ( ) ( V11() V12() real ) Real) / x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) & x0 : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) & x1 : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) & x2 : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) & x3 : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) & x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) are_mutually_different holds
[!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = - (k : ( ( ) ( V11() V12() real ) Real) / (((x0 : ( ( ) ( V11() V12() real ) Real) * x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * x2 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * x3 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:37
for k, x0, x1, x2, x3, x4 being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st ( for x being ( ( ) ( V11() V12() real ) Real) holds f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = k : ( ( ) ( V11() V12() real ) Real) / x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) & x0 : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) & x1 : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) & x2 : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) & x3 : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) & x4 : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) & x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) are_mutually_different holds
[!f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) ,x2 : ( ( ) ( V11() V12() real ) Real) ,x3 : ( ( ) ( V11() V12() real ) Real) ,x4 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) Real) = k : ( ( ) ( V11() V12() real ) Real) / ((((x0 : ( ( ) ( V11() V12() real ) Real) * x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * x2 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * x3 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * x4 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:38
for k, h being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st ( for x being ( ( ) ( V11() V12() real ) Real) holds
( f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = k : ( ( ) ( V11() V12() real ) Real) / x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) & x : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) & x : ( ( ) ( V11() V12() real ) Real) + h : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) ) holds
for x being ( ( ) ( V11() V12() real ) Real) holds (fD (f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (- (k : ( ( ) ( V11() V12() real ) Real) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / ((x : ( ( ) ( V11() V12() real ) Real) + h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:39
for k, h being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st ( for x being ( ( ) ( V11() V12() real ) Real) holds
( f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = k : ( ( ) ( V11() V12() real ) Real) / x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) & x : ( ( ) ( V11() V12() real ) Real) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) & x : ( ( ) ( V11() V12() real ) Real) - h : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) ) holds
for x being ( ( ) ( V11() V12() real ) Real) holds (bD (f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (- (k : ( ( ) ( V11() V12() real ) Real) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / ((x : ( ( ) ( V11() V12() real ) Real) - h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:40
for k, h being ( ( ) ( V11() V12() real ) Real)
for f being ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) st ( for x being ( ( ) ( V11() V12() real ) Real) holds
( f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = k : ( ( ) ( V11() V12() real ) Real) / x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) & x : ( ( ) ( V11() V12() real ) Real) + (h : ( ( ) ( V11() V12() real ) Real) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) & x : ( ( ) ( V11() V12() real ) Real) - (h : ( ( ) ( V11() V12() real ) Real) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) <> 0 : ( ( ) ( natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) ) holds
for x being ( ( ) ( V11() V12() real ) Real) holds (cD (f : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (- (k : ( ( ) ( V11() V12() real ) Real) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / ((x : ( ( ) ( V11() V12() real ) Real) - (h : ( ( ) ( V11() V12() real ) Real) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (x : ( ( ) ( V11() V12() real ) Real) + (h : ( ( ) ( V11() V12() real ) Real) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:41
for x0, x1 being ( ( ) ( V11() V12() real ) Real) holds [!sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = ((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * (cos ((x0 : ( ( ) ( V11() V12() real ) Real) + x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin ((x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / (x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:42
for h, x being ( ( ) ( V11() V12() real ) Real) holds (fD (sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * ((cos (((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin (h : ( ( ) ( V11() V12() real ) Real) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:43
for h, x being ( ( ) ( V11() V12() real ) Real) holds (bD (sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * ((cos (((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin (h : ( ( ) ( V11() V12() real ) Real) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:44
for h, x being ( ( ) ( V11() V12() real ) Real) holds (cD (sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * ((cos x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin (h : ( ( ) ( V11() V12() real ) Real) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:45
for x0, x1 being ( ( ) ( V11() V12() real ) Real) holds [!cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = - (((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * (sin ((x0 : ( ( ) ( V11() V12() real ) Real) + x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin ((x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / (x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:46
for h, x being ( ( ) ( V11() V12() real ) Real) holds (fD (cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = - (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * ((sin (((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin (h : ( ( ) ( V11() V12() real ) Real) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:47
for h, x being ( ( ) ( V11() V12() real ) Real) holds (bD (cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = - (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * ((sin (((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin (h : ( ( ) ( V11() V12() real ) Real) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:48
for h, x being ( ( ) ( V11() V12() real ) Real) holds (cD (cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = - (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * ((sin x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin (h : ( ( ) ( V11() V12() real ) Real) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:49
for x0, x1 being ( ( ) ( V11() V12() real ) Real) holds [!(sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = ((1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * ((cos (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - (cos (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x0 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / (x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:50
for h, x being ( ( ) ( V11() V12() real ) Real) holds (fD ((sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * ((cos (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - (cos (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * (x : ( ( ) ( V11() V12() real ) Real) + h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:51
for h, x being ( ( ) ( V11() V12() real ) Real) holds (bD ((sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * ((cos (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * (x : ( ( ) ( V11() V12() real ) Real) - h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - (cos (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:52
for h, x being ( ( ) ( V11() V12() real ) Real) holds (cD ((sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * ((cos ((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - (cos ((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:53
for x0, x1 being ( ( ) ( V11() V12() real ) Real) holds [!(sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = ((1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * ((sin (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x0 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - (sin (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / (x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:54
for h, x being ( ( ) ( V11() V12() real ) Real) holds (fD ((sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * ((sin (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * (x : ( ( ) ( V11() V12() real ) Real) + h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - (sin (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:55
for h, x being ( ( ) ( V11() V12() real ) Real) holds (bD ((sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * ((sin (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - (sin (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * (x : ( ( ) ( V11() V12() real ) Real) - h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:56
for h, x being ( ( ) ( V11() V12() real ) Real) holds (cD ((sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * ((sin ((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - (sin ((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:57
for x0, x1 being ( ( ) ( V11() V12() real ) Real) holds [!(cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = ((1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * ((cos (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x0 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - (cos (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / (x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:58
for h, x being ( ( ) ( V11() V12() real ) Real) holds (fD ((cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * ((cos (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * (x : ( ( ) ( V11() V12() real ) Real) + h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - (cos (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:59
for h, x being ( ( ) ( V11() V12() real ) Real) holds (bD ((cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * ((cos (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - (cos (2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * (x : ( ( ) ( V11() V12() real ) Real) - h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:60
for h, x being ( ( ) ( V11() V12() real ) Real) holds (cD ((cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * ((cos ((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - (cos ((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:61
for x0, x1 being ( ( ) ( V11() V12() real ) Real) holds [!((sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = - (((1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (((sin ((3 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * (x1 : ( ( ) ( V11() V12() real ) Real) + x0 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin ((3 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * (x1 : ( ( ) ( V11() V12() real ) Real) - x0 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + ((sin ((x0 : ( ( ) ( V11() V12() real ) Real) + x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin ((x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / (x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:62
for h, x being ( ( ) ( V11() V12() real ) Real) holds (fD (((sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (((sin (((6 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + (3 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin ((3 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - ((sin (((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin (h : ( ( ) ( V11() V12() real ) Real) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:63
for h, x being ( ( ) ( V11() V12() real ) Real) holds (bD (((sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = ((1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * ((sin (((6 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - (3 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin ((3 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - ((1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * ((sin (((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin (h : ( ( ) ( V11() V12() real ) Real) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:64
for h, x being ( ( ) ( V11() V12() real ) Real) holds (cD (((sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (- ((1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * ((sin x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin (h : ( ( ) ( V11() V12() real ) Real) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + ((1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * ((sin (3 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin ((3 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:65
for x0, x1 being ( ( ) ( V11() V12() real ) Real) holds [!((sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = ((1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (((cos ((x0 : ( ( ) ( V11() V12() real ) Real) + x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin ((x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + ((cos ((3 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * (x0 : ( ( ) ( V11() V12() real ) Real) + x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin ((3 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * (x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / (x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:66
for h, x being ( ( ) ( V11() V12() real ) Real) holds (fD (((sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (((cos (((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin (h : ( ( ) ( V11() V12() real ) Real) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + ((cos (((6 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + (3 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin ((3 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:67
for h, x being ( ( ) ( V11() V12() real ) Real) holds (bD (((sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (((cos (((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin (h : ( ( ) ( V11() V12() real ) Real) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + ((cos (((6 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) - (3 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin ((3 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:68
for h, x being ( ( ) ( V11() V12() real ) Real) holds (cD (((sin : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like ) ( V1() Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like V26( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) V30( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ) complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,h : ( ( ) ( V11() V12() real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( V11() V12() real ) Real) : ( ( ) ( V11() V12() real ) set ) = (1 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (((cos x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin (h : ( ( ) ( V11() V12() real ) Real) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) + ((cos (3 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * x : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin ((3 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * h : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:69
for x0, x1 being ( ( ) ( V11() V12() real ) Real) st x0 : ( ( ) ( V11() V12() real ) Real) in dom tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) & x1 : ( ( ) ( V11() V12() real ) Real) in dom tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) holds
[!tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = (sin (x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / (((cos x0 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (cos x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:70
for x0, x1 being ( ( ) ( V11() V12() real ) Real) st x0 : ( ( ) ( V11() V12() real ) Real) in dom cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) & x1 : ( ( ) ( V11() V12() real ) Real) in dom cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) holds
[!cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = - ((sin (x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / (((sin x0 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:71
for x0, x1 being ( ( ) ( V11() V12() real ) Real) st x0 : ( ( ) ( V11() V12() real ) Real) in dom cosec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) & x1 : ( ( ) ( V11() V12() real ) Real) in dom cosec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) holds
[!cosec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = ((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * (cos ((x1 : ( ( ) ( V11() V12() real ) Real) + x0 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin ((x1 : ( ( ) ( V11() V12() real ) Real) - x0 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / (((sin x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin x0 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;

theorem :: DIFF_2:72
for x0, x1 being ( ( ) ( V11() V12() real ) Real) st x0 : ( ( ) ( V11() V12() real ) Real) in dom sec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) & x1 : ( ( ) ( V11() V12() real ) Real) in dom sec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) holds
[!sec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -defined REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) M2( bool [:REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) ,REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) )) ,x0 : ( ( ) ( V11() V12() real ) Real) ,x1 : ( ( ) ( V11() V12() real ) Real) !] : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) = - (((2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) * (sin ((x1 : ( ( ) ( V11() V12() real ) Real) + x0 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (sin ((x1 : ( ( ) ( V11() V12() real ) Real) - x0 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / 2 : ( ( ) ( V1() natural V11() V12() real V44() V45() V46() V47() V48() V49() V57() V58() ) M3( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) , NAT : ( ( ) ( V44() V45() V46() V47() V48() V49() V50() ) M2( bool REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) : ( ( ) ( ) set ) )) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) / (((cos x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (cos x0 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) * (x0 : ( ( ) ( V11() V12() real ) Real) - x1 : ( ( ) ( V11() V12() real ) Real) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) : ( ( ) ( V11() V12() real ) M2( REAL : ( ( ) ( V1() V44() V45() V46() V50() V59() ) set ) )) ;