begin
theorem
for
k,
x0,
x1,
x2,
x3 being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) st ( for
x being ( ( ) (
V22()
real ext-real )
Real) holds
f : ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= k : ( ( ) (
V22()
real ext-real )
Real)
/ (x : ( ( ) ( V22() real ext-real ) Real) ^2) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
COMPLEX : ( ( ) ( non
empty V35()
V57()
V63() )
set ) ) ) &
x0 : ( ( ) (
V22()
real ext-real )
Real)
<> 0 : ( ( ) (
Relation-like non-empty empty-yielding RAT : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V60()
V63() )
set )
-valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22()
real ext-real non
positive non
negative V33()
V46()
V47()
V48()
V49()
V56()
V57()
V58()
V59()
V60()
V61()
V62()
V63()
V88()
V89()
V90()
V91()
V92()
V93()
V94()
V95()
V96()
V97()
V98()
V99() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) ) &
x1 : ( ( ) (
V22()
real ext-real )
Real)
<> 0 : ( ( ) (
Relation-like non-empty empty-yielding RAT : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V60()
V63() )
set )
-valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22()
real ext-real non
positive non
negative V33()
V46()
V47()
V48()
V49()
V56()
V57()
V58()
V59()
V60()
V61()
V62()
V63()
V88()
V89()
V90()
V91()
V92()
V93()
V94()
V95()
V96()
V97()
V98()
V99() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) ) &
x2 : ( ( ) (
V22()
real ext-real )
Real)
<> 0 : ( ( ) (
Relation-like non-empty empty-yielding RAT : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V60()
V63() )
set )
-valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22()
real ext-real non
positive non
negative V33()
V46()
V47()
V48()
V49()
V56()
V57()
V58()
V59()
V60()
V61()
V62()
V63()
V88()
V89()
V90()
V91()
V92()
V93()
V94()
V95()
V96()
V97()
V98()
V99() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) ) &
x3 : ( ( ) (
V22()
real ext-real )
Real)
<> 0 : ( ( ) (
Relation-like non-empty empty-yielding RAT : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V60()
V63() )
set )
-valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22()
real ext-real non
positive non
negative V33()
V46()
V47()
V48()
V49()
V56()
V57()
V58()
V59()
V60()
V61()
V62()
V63()
V88()
V89()
V90()
V91()
V92()
V93()
V94()
V95()
V96()
V97()
V98()
V99() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) ) &
x0 : ( ( ) (
V22()
real ext-real )
Real) ,
x1 : ( ( ) (
V22()
real ext-real )
Real) ,
x2 : ( ( ) (
V22()
real ext-real )
Real) ,
x3 : ( ( ) (
V22()
real ext-real )
Real)
are_mutually_different holds
[!f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,x0 : ( ( ) ( V22() real ext-real ) Real) ,x1 : ( ( ) ( V22() real ext-real ) Real) ,x2 : ( ( ) ( V22() real ext-real ) Real) ,x3 : ( ( ) ( V22() real ext-real ) Real) !] : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (k : ( ( ) ( V22() real ext-real ) Real) * (((1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) / ((x1 : ( ( ) ( V22() real ext-real ) Real) * x2 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * x0 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) * (((1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) / x0 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) + (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) / x2 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) + (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) / x1 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) - ((1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) / ((x2 : ( ( ) ( V22() real ext-real ) Real) * x1 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * x3 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) * (((1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) / x3 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) + (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) / x1 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) + (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) / x2 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
/ (x0 : ( ( ) ( V22() real ext-real ) Real) - x3 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
COMPLEX : ( ( ) ( non
empty V35()
V57()
V63() )
set ) ) ;
theorem
for
x0,
x1 being ( ( ) (
V22()
real ext-real )
Real) st
x0 : ( ( ) (
V22()
real ext-real )
Real)
in (dom cosec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) )
/\ (dom sec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x1 : ( ( ) (
V22()
real ext-real )
Real)
in (dom cosec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) )
/\ (dom sec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
[!(cosec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V22() real ext-real ) Real) ,x1 : ( ( ) ( V22() real ext-real ) Real) !] : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * ((cos (x1 : ( ( ) ( V22() real ext-real ) Real) + x0 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (sin (x1 : ( ( ) ( V22() real ext-real ) Real) - x0 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) / ((sin (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * x0 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (sin (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * x1 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
COMPLEX : ( ( ) ( non
empty V35()
V57()
V63() )
set ) )
/ (x0 : ( ( ) ( V22() real ext-real ) Real) - x1 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
COMPLEX : ( ( ) ( non
empty V35()
V57()
V63() )
set ) ) ;
theorem
for
x,
h being ( ( ) (
V22()
real ext-real )
Real) st
x : ( ( ) (
V22()
real ext-real )
Real)
+ h : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in (dom cosec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) )
/\ (dom sec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x : ( ( ) (
V22()
real ext-real )
Real)
in (dom cosec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) )
/\ (dom sec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
(fD ((cosec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= - (4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * (((cos ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) + h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (sin h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) / ((sin (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * (x : ( ( ) ( V22() real ext-real ) Real) + h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (sin (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
x,
h being ( ( ) (
V22()
real ext-real )
Real) st
x : ( ( ) (
V22()
real ext-real )
Real)
- h : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in (dom cosec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) )
/\ (dom sec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x : ( ( ) (
V22()
real ext-real )
Real)
in (dom cosec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) )
/\ (dom sec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
(bD ((cosec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= - (4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * (((cos ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) - h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (sin h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) / ((sin (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (sin (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * (x : ( ( ) ( V22() real ext-real ) Real) - h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
x,
h being ( ( ) (
V22()
real ext-real )
Real) st
x : ( ( ) (
V22()
real ext-real )
Real)
+ (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
COMPLEX : ( ( ) ( non
empty V35()
V57()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in (dom cosec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) )
/\ (dom sec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x : ( ( ) (
V22()
real ext-real )
Real)
- (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
COMPLEX : ( ( ) ( non
empty V35()
V57()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in (dom cosec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) )
/\ (dom sec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
(cD ((cosec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sec : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= - (4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * (((cos (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (sin h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) / ((sin ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) + h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (sin ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) - h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
x0,
x1 being ( ( ) (
V22()
real ext-real )
Real) st
x0 : ( ( ) (
V22()
real ext-real )
Real)
in dom tan : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x1 : ( ( ) (
V22()
real ext-real )
Real)
in dom tan : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
[!((tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V22() real ext-real ) Real) ,x1 : ( ( ) ( V22() real ext-real ) Real) !] : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= [!(tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V22() real ext-real ) Real) ,x1 : ( ( ) ( V22() real ext-real ) Real) !] : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
x,
h being ( ( ) (
V22()
real ext-real )
Real) st
x : ( ( ) (
V22()
real ext-real )
Real)
in dom tan : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x : ( ( ) (
V22()
real ext-real )
Real)
+ h : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in dom tan : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
(fD (((tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- ((tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
x,
h being ( ( ) (
V22()
real ext-real )
Real) st
x : ( ( ) (
V22()
real ext-real )
Real)
in dom tan : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x : ( ( ) (
V22()
real ext-real )
Real)
- h : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in dom tan : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
(bD (((tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- ((tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
x,
h being ( ( ) (
V22()
real ext-real )
Real) st
x : ( ( ) (
V22()
real ext-real )
Real)
+ (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
COMPLEX : ( ( ) ( non
empty V35()
V57()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in dom tan : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x : ( ( ) (
V22()
real ext-real )
Real)
- (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
COMPLEX : ( ( ) ( non
empty V35()
V57()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in dom tan : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
(cD (((tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- ((tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
x0,
x1 being ( ( ) (
V22()
real ext-real )
Real) st
x0 : ( ( ) (
V22()
real ext-real )
Real)
in dom cot : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x1 : ( ( ) (
V22()
real ext-real )
Real)
in dom cot : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
[!((cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V22() real ext-real ) Real) ,x1 : ( ( ) ( V22() real ext-real ) Real) !] : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= [!(cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V22() real ext-real ) Real) ,x1 : ( ( ) ( V22() real ext-real ) Real) !] : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
x,
h being ( ( ) (
V22()
real ext-real )
Real) st
x : ( ( ) (
V22()
real ext-real )
Real)
in dom cot : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x : ( ( ) (
V22()
real ext-real )
Real)
+ h : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in dom cot : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
(fD (((cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- ((cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
x,
h being ( ( ) (
V22()
real ext-real )
Real) st
x : ( ( ) (
V22()
real ext-real )
Real)
in dom cot : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x : ( ( ) (
V22()
real ext-real )
Real)
- h : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in dom cot : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
(bD (((cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- ((cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
x,
h being ( ( ) (
V22()
real ext-real )
Real) st
x : ( ( ) (
V22()
real ext-real )
Real)
+ (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
COMPLEX : ( ( ) ( non
empty V35()
V57()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in dom cot : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x : ( ( ) (
V22()
real ext-real )
Real)
- (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
COMPLEX : ( ( ) ( non
empty V35()
V57()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in dom cot : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
(cD (((cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- ((cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
(fD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((Shift (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x0,
x1 being ( ( ) (
V22()
real ext-real )
Real)
for
f,
g being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) st ( for
x being ( ( ) (
V22()
real ext-real )
Real) holds
f : ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (fD (g : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) holds
[!f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,x0 : ( ( ) ( V22() real ext-real ) Real) ,x1 : ( ( ) ( V22() real ext-real ) Real) !] : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= [!g : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,(x0 : ( ( ) ( V22() real ext-real ) Real) + h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,(x1 : ( ( ) ( V22() real ext-real ) Real) + h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) !] : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- [!g : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,x0 : ( ( ) ( V22() real ext-real ) Real) ,x1 : ( ( ) ( V22() real ext-real ) Real) !] : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
(fD ((fD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((fD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * ((fD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
(bD ((fD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((fD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- ((bD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
(cD ((fD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((fD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- ((cD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
((fdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (((fdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V35() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() real ext-real non positive non negative V33() V46() V47() V48() V49() V56() V57() V58() V59() V60() V61() V62() V63() V88() V89() V90() V91() V92() V93() V94() V95() V96() V97() V98() V99() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- (((fdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V35() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() real ext-real non positive non negative V33() V46() V47() V48() V49() V56() V57() V58() V59() V60() V61() V62() V63() V88() V89() V90() V91() V92() V93() V94() V95() V96() V97() V98() V99() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V22()
real ext-real V33()
V56()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) )
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
((fdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() real ext-real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() real ext-real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (((fdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() real ext-real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- (((fdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() real ext-real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
(bD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- ((Shift (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,(- h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) )) : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x0,
x1 being ( ( ) (
V22()
real ext-real )
Real)
for
f,
g being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) st ( for
x being ( ( ) (
V22()
real ext-real )
Real) holds
f : ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (bD (g : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) holds
[!f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,x0 : ( ( ) ( V22() real ext-real ) Real) ,x1 : ( ( ) ( V22() real ext-real ) Real) !] : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= [!g : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,x0 : ( ( ) ( V22() real ext-real ) Real) ,x1 : ( ( ) ( V22() real ext-real ) Real) !] : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- [!g : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,(x0 : ( ( ) ( V22() real ext-real ) Real) - h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,(x1 : ( ( ) ( V22() real ext-real ) Real) - h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) !] : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
(fD ((bD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((fD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- ((bD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
(bD ((bD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * ((bD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- ((bD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) * h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
(cD ((bD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((cD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- ((bD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
((bdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (((bdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V35() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() real ext-real non positive non negative V33() V46() V47() V48() V49() V56() V57() V58() V59() V60() V61() V62() V63() V88() V89() V90() V91() V92() V93() V94() V95() V96() V97() V98() V99() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- (((bdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V35() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() real ext-real non positive non negative V33() V46() V47() V48() V49() V56() V57() V58() V59() V60() V61() V62() V63() V88() V89() V90() V91() V92() V93() V94() V95() V96() V97() V98() V99() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V22()
real ext-real V33()
V56()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) )
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
((bdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() real ext-real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() real ext-real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (((bdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() real ext-real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- (((bdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() real ext-real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
(cD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((Shift (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,(h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) )) : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- ((Shift (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,(- (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) )) : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x0,
x1 being ( ( ) (
V22()
real ext-real )
Real)
for
f,
g being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) st ( for
x being ( ( ) (
V22()
real ext-real )
Real) holds
f : ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (cD (g : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) holds
[!f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,x0 : ( ( ) ( V22() real ext-real ) Real) ,x1 : ( ( ) ( V22() real ext-real ) Real) !] : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= [!g : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,(x0 : ( ( ) ( V22() real ext-real ) Real) + (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,(x1 : ( ( ) ( V22() real ext-real ) Real) + (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) !] : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- [!g : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,(x0 : ( ( ) ( V22() real ext-real ) Real) - (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,(x1 : ( ( ) ( V22() real ext-real ) Real) - (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) !] : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
(fD ((cD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((fD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- ((cD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
(bD ((cD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((cD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- ((bD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
(cD ((cD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((fD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- ((bD (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
((cdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (((cdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V35() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() real ext-real non positive non negative V33() V46() V47() V48() V49() V56() V57() V58() V59() V60() V61() V62() V63() V88() V89() V90() V91() V92() V93() V94() V95() V96() V97() V98() V99() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- (((cdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V35() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() real ext-real non positive non negative V33() V46() V47() V48() V49() V56() V57() V58() V59() V60() V61() V62() V63() V88() V89() V90() V91() V92() V93() V94() V95() V96() V97() V98() V99() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V22()
real ext-real V33()
V56()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) )
for
h,
x being ( ( ) (
V22()
real ext-real )
Real)
for
f being ( (
Function-like V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like non
empty V14(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V30(
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
V46()
V47()
V48() )
Function of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ,
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) holds
((cdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() real ext-real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() real ext-real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (((cdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() real ext-real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- (((cdif (f : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Function of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) -defined K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) ) V94() V95() V96() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ,K36(REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) : ( ( ) ( functional non empty V88() V89() V90() ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() real ext-real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
x0,
x1 being ( ( ) (
V22()
real ext-real )
Real) st
x0 : ( ( ) (
V22()
real ext-real )
Real)
in dom tan : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x1 : ( ( ) (
V22()
real ext-real )
Real)
in dom tan : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
[!((tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V22() real ext-real ) Real) ,x1 : ( ( ) ( V22() real ext-real ) Real) !] : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((((sin x0 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * ((cos x1 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ^2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) - (((sin x1 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * ((cos x0 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ^2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
/ ((((cos x0 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ^2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * ((cos x1 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ^2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (x0 : ( ( ) ( V22() real ext-real ) Real) - x1 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
COMPLEX : ( ( ) ( non
empty V35()
V57()
V63() )
set ) ) ;
theorem
for
x,
h being ( ( ) (
V22()
real ext-real )
Real) st
x : ( ( ) (
V22()
real ext-real )
Real)
in dom tan : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x : ( ( ) (
V22()
real ext-real )
Real)
+ h : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in dom tan : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
(fD (((tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (((sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (((cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ") : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- (((sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (((cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ") : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
x,
h being ( ( ) (
V22()
real ext-real )
Real) st
x : ( ( ) (
V22()
real ext-real )
Real)
in dom tan : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x : ( ( ) (
V22()
real ext-real )
Real)
- h : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in dom tan : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
(bD (((tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (((sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (((cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ") : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- (((sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (((cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ") : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
x,
h being ( ( ) (
V22()
real ext-real )
Real) st
x : ( ( ) (
V22()
real ext-real )
Real)
+ (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
COMPLEX : ( ( ) ( non
empty V35()
V57()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in dom tan : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x : ( ( ) (
V22()
real ext-real )
Real)
- (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
COMPLEX : ( ( ) ( non
empty V35()
V57()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in dom tan : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
(cD (((tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) tan : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (((sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (((cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ") : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- (((sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (((cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ") : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
x0,
x1 being ( ( ) (
V22()
real ext-real )
Real) st
x0 : ( ( ) (
V22()
real ext-real )
Real)
in dom cot : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x1 : ( ( ) (
V22()
real ext-real )
Real)
in dom cot : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
[!((cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V22() real ext-real ) Real) ,x1 : ( ( ) ( V22() real ext-real ) Real) !] : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= ((((cos x0 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * ((sin x1 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ^2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) - (((cos x1 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * ((sin x0 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ^2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
/ ((((sin x0 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ^2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * ((sin x1 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ^2) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (x0 : ( ( ) ( V22() real ext-real ) Real) - x1 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
COMPLEX : ( ( ) ( non
empty V35()
V57()
V63() )
set ) ) ;
theorem
for
x,
h being ( ( ) (
V22()
real ext-real )
Real) st
x : ( ( ) (
V22()
real ext-real )
Real)
in dom cot : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x : ( ( ) (
V22()
real ext-real )
Real)
+ h : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in dom cot : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
(fD (((cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (((cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (((sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ") : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- (((cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (((sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ") : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
x,
h being ( ( ) (
V22()
real ext-real )
Real) st
x : ( ( ) (
V22()
real ext-real )
Real)
in dom cot : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x : ( ( ) (
V22()
real ext-real )
Real)
- h : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in dom cot : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
(bD (((cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (((cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (((sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ") : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- (((cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (((sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - h : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ") : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
x,
h being ( ( ) (
V22()
real ext-real )
Real) st
x : ( ( ) (
V22()
real ext-real )
Real)
+ (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
COMPLEX : ( ( ) ( non
empty V35()
V57()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in dom cot : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) &
x : ( ( ) (
V22()
real ext-real )
Real)
- (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
COMPLEX : ( ( ) ( non
empty V35()
V57()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
in dom cot : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) ) : ( ( ) (
V57()
V58()
V59() )
Element of
bool REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) : ( ( ) ( )
set ) ) holds
(cD (((cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cot : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) (#) cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) ,h : ( ( ) ( V22() real ext-real ) Real) )) : ( (
Function-like ) (
Relation-like REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-defined REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set )
-valued Function-like V46()
V47()
V48() )
Element of
bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) (
Relation-like V46()
V47()
V48() )
set ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
real ext-real )
Real) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
= (((cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (((sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) + (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ") : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) )
- (((cos : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (((sin : ( ( Function-like V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -defined REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) -valued Function-like non empty V14( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V30( REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) , REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) V46() V47() V48() ) Element of bool [:REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ,REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) :] : ( ( ) ( Relation-like V46() V47() V48() ) set ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() real ext-real ) Real) - (h : ( ( ) ( V22() real ext-real ) Real) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ") : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) |^ 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() ) Element of bool REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V22()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V35()
V57()
V58()
V59()
V63() )
set ) ) ;