begin
theorem
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
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
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
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
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
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
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
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
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
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
([!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
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
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
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
([!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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 ) )) ;