:: DIFF_4 semantic presentation

begin

theorem :: DIFF_4:1
for x0, x1 being ( ( ) ( V22() real ext-real ) Real) st 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 ) ) ) holds
(log (number_e : ( ( ) ( 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 ) ) - (log (number_e : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) 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 ) ) = log (number_e : ( ( ) ( 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 COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ) : ( ( real ) ( V22() real ext-real ) set ) ;

theorem :: DIFF_4:2
for x0, x1 being ( ( ) ( V22() real ext-real ) Real) st 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 ) ) ) holds
(log (number_e : ( ( ) ( 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 ) ) + (log (number_e : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) 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 ) ) = log (number_e : ( ( ) ( 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 ) ) ;

theorem :: DIFF_4:3
for x being ( ( ) ( V22() real ext-real ) Real) st x : ( ( ) ( 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 ) ) ) holds
log (number_e : ( ( ) ( V22() real ext-real ) Element of 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 ) ) = ln : ( ( 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 ) ) ;

theorem :: DIFF_4:4
for x0, x1 being ( ( ) ( V22() real ext-real ) Real) st 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 ) ) ) holds
(ln : ( ( 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) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) - (ln : ( ( 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 ) ) . 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 ) ) = ln : ( ( 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 COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) ;

theorem :: DIFF_4:5
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 :: DIFF_4:6
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 ) ) ,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 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 ) ) - ((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 ) ) / ((((sin x0 : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( 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 ) ) ) : ( ( ) ( 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 ) ) * (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 ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ;

theorem :: DIFF_4:7
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 ) ) ,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 ) ) = ((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 ) ) ) / 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 ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) 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) + 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 ) ) - (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 ) ) ) : ( ( ) ( 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 (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 ) ) * (sin 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) : ( ( ) ( 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 :: DIFF_4:8
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 ) ) ,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 ) ) = ((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 ) ) ) / 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 ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) 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 ) ) - (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 ) ) ) * (h : ( ( ) ( V22() real ext-real ) Real) - 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 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 x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( V22() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V57() V58() V59() V63() ) set ) ) * (sin (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) : ( ( ) ( 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 :: DIFF_4:9
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 ) ) ,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 ) ) = ((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 ) ) ) / 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 ) ) ) ) : ( ( ) ( non empty V22() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) * ((cos (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 ) ) ) * 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 ) ) - (cos (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 ) ) ) * 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 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 (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 ) ) * (sin (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) : ( ( ) ( 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 :: DIFF_4:10
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 ) ) & 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 ) ) 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 ) ) (#) 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 ) ) ) : ( ( 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 ) ) ) * ((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 ) ) * (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 ) ) / ((((cos (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 ) ) - (cos (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 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 ) ) * (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 :: DIFF_4:11
for x, h being ( ( ) ( V22() real ext-real ) Real) st 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 ) ) & 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 ) ) 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 ) ) (#) 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 ) ) ) : ( ( 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 ) ) ) * (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 ) ) * (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 ) ) / (((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 ) ) - (cos 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 ) ) ^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 ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ;

theorem :: DIFF_4:12
for x, h being ( ( ) ( V22() real ext-real ) Real) st 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 ) ) & 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 ) ) 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 ) ) (#) 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 ) ) ) : ( ( 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 ) ) ) * (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 ) ) * (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 ) ) / (((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 ) ) - (cos 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 ) ) ^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 ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ;

theorem :: DIFF_4:13
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 ) ) & 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 ) ) 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 ) ) (#) 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 ) ) ) : ( ( 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 ) ) ) * (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 ) ) * (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 ) ) / (((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 ) ) - (cos 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 ) ) ^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 ) ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ;

theorem :: DIFF_4:14
for x0, x1 being ( ( ) ( V22() real ext-real ) Real) st x0 : ( ( ) ( V22() real ext-real ) Real) in 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 ) ) & x1 : ( ( ) ( V22() real ext-real ) Real) in 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 ) ) holds
[!(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 ) ) (#) 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 ) ) ) * ((sin (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 ) ) * (sin (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 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) + 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 ) ) + (cos (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 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 ) ) * (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 :: DIFF_4:15
for x, h being ( ( ) ( V22() real ext-real ) Real) st x : ( ( ) ( V22() real ext-real ) Real) in 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 ) ) & 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 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 ) ) holds
(fD ((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 ) ) (#) 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 ) ) ) * (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 ) ) * (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 ) ) / (((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 ) ) + (cos 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 ) ) ^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 ) ) ;

theorem :: DIFF_4:16
for x, h being ( ( ) ( V22() real ext-real ) Real) st x : ( ( ) ( V22() real ext-real ) Real) in 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 ) ) & 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 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 ) ) holds
(bD ((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 ) ) (#) 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 ) ) ) * (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 ) ) * (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 ) ) / (((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 ) ) + (cos 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 ) ) ^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 ) ) ;

theorem :: DIFF_4:17
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 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 ) ) & 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 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 ) ) holds
(cD ((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 ) ) (#) 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 ) ) ) * (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 ) ) * (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 ) ) / (((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 ) ) + (cos 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 ) ) ^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 ) ) ;

theorem :: DIFF_4:18
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 :: DIFF_4:19
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 :: DIFF_4:20
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 :: DIFF_4:21
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 :: DIFF_4:22
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 :: DIFF_4:23
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 :: DIFF_4:24
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 :: DIFF_4:25
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 :: DIFF_4:26
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 :: DIFF_4:27
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 :: DIFF_4:28
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 :: DIFF_4:29
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 :: DIFF_4:30
for x0, x1 being ( ( ) ( V22() real ext-real ) Real) st 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 ) ) ) holds
[!ln : ( ( 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 ) ) = (ln : ( ( 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 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) - 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 :: DIFF_4:31
for x, h being ( ( ) ( V22() real ext-real ) Real) st x : ( ( ) ( 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 ) ) ) & 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 ) ) > 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 ) ) ) holds
(fD (ln : ( ( 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 ) ) = ln : ( ( 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 ) ) . (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 ) ) ) + (h : ( ( ) ( V22() real ext-real ) Real) / x : ( ( ) ( V22() real ext-real ) Real) ) : ( ( ) ( 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 :: DIFF_4:32
for x, h being ( ( ) ( V22() real ext-real ) Real) st x : ( ( ) ( 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 ) ) ) & 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 ) ) > 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 ) ) ) holds
(bD (ln : ( ( 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 ) ) = ln : ( ( 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 ) ) . (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 ) ) ) + (h : ( ( ) ( V22() real ext-real ) Real) / (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 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 :: DIFF_4:33
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 ) ) > 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 ) ) ) & 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 ) ) > 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 ) ) ) holds
(cD (ln : ( ( 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 ) ) = ln : ( ( 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 ) ) . (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 ) ) ) + (h : ( ( ) ( V22() real ext-real ) Real) / (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 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 :: DIFF_4:34
for h, k being ( ( real ) ( V22() real ext-real ) number ) holds exp_R (h : ( ( real ) ( V22() real ext-real ) number ) - k : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) set ) : ( ( ) ( V22() real ext-real ) set ) = (exp_R h : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) set ) / (exp_R k : ( ( real ) ( V22() real ext-real ) number ) ) : ( ( ) ( V22() real ext-real ) set ) : ( ( ) ( V22() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V35() V57() V63() ) set ) ) ;

theorem :: DIFF_4:35
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 :: DIFF_4:36
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 :: DIFF_4:37
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 :: DIFF_4:38
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 :: DIFF_4:39
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 :: DIFF_4:40
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 :: DIFF_4:41
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 :: DIFF_4:42
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 :: DIFF_4:43
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 :: DIFF_4:44
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 :: DIFF_4:45
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 :: DIFF_4:46
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 :: DIFF_4:47
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 :: DIFF_4:48
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 :: DIFF_4:49
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 :: DIFF_4:50
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 :: DIFF_4:51
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 :: DIFF_4:52
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 :: DIFF_4:53
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 :: DIFF_4:54
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 :: DIFF_4:55
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 :: DIFF_4:56
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 :: DIFF_4:57
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 :: DIFF_4:58
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 :: DIFF_4:59
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 :: DIFF_4:60
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 :: DIFF_4:61
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 :: DIFF_4:62
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 :: DIFF_4:63
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 ) ) ;