:: LIMFUNC3 semantic presentation

begin

theorem :: LIMFUNC3:1
for x0 being ( ( ) ( V11() real ext-real ) Real)
for seq being ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st ( rng seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V69() V70() V71() ) set ) c= (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (left_open_halfline x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) or rng seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V69() V70() V71() ) set ) c= (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (right_open_halfline x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds
rng seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V69() V70() V71() ) set ) c= (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) \ {x0 : ( ( ) ( V11() real ext-real ) Real) } : ( ( ) ( V69() V70() V71() ) set ) : ( ( ) ( ) Element of K6((dom b3 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: LIMFUNC3:2
for x0 being ( ( ) ( V11() real ext-real ) Real)
for seq being ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < abs (x0 : ( ( ) ( V11() real ext-real ) Real) - (seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) & abs (x0 : ( ( ) ( V11() real ext-real ) Real) - (seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) < 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) / (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) & seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) ) ) holds
( seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) = x0 : ( ( ) ( V11() real ext-real ) Real) & rng seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V69() V70() V71() ) set ) c= dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & rng seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V69() V70() V71() ) set ) c= (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) \ {x0 : ( ( ) ( V11() real ext-real ) Real) } : ( ( ) ( V69() V70() V71() ) set ) : ( ( ) ( ) Element of K6((dom b3 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: LIMFUNC3:3
for x0 being ( ( ) ( V11() real ext-real ) Real)
for seq being ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) = x0 : ( ( ) ( V11() real ext-real ) Real) & rng seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V69() V70() V71() ) set ) c= (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) \ {x0 : ( ( ) ( V11() real ext-real ) Real) } : ( ( ) ( V69() V70() V71() ) set ) : ( ( ) ( ) Element of K6((dom b3 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) st
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < abs (x0 : ( ( ) ( V11() real ext-real ) Real) - (seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) & abs (x0 : ( ( ) ( V11() real ext-real ) Real) - (seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) & seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) ) ;

theorem :: LIMFUNC3:4
for r, x0 being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \ {x0 : ( ( ) ( V11() real ext-real ) Real) } : ( ( ) ( V69() V70() V71() ) set ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) = ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: LIMFUNC3:5
for r2, x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r2 : ( ( ) ( V11() real ext-real ) Real) & ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r2 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r2 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) holds
for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) ) ;

theorem :: LIMFUNC3:6
for x0 being ( ( ) ( V11() real ext-real ) Real)
for seq being ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( x0 : ( ( ) ( V11() real ext-real ) Real) - (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) / (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) < seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) & seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) < x0 : ( ( ) ( V11() real ext-real ) Real) & seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) ) ) holds
( seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) = x0 : ( ( ) ( V11() real ext-real ) Real) & rng seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V69() V70() V71() ) set ) c= (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) \ {x0 : ( ( ) ( V11() real ext-real ) Real) } : ( ( ) ( V69() V70() V71() ) set ) : ( ( ) ( ) Element of K6((dom b3 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: LIMFUNC3:7
for x0, g being ( ( ) ( V11() real ext-real ) Real)
for seq being ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) = x0 : ( ( ) ( V11() real ext-real ) Real) & 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < g : ( ( ) ( V11() real ext-real ) Real) holds
ex k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) st k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( x0 : ( ( ) ( V11() real ext-real ) Real) - g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) < seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) & seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) < x0 : ( ( ) ( V11() real ext-real ) Real) + g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ;

theorem :: LIMFUNC3:8
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) holds
( ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) ) ) iff ( ( for r being ( ( ) ( V11() real ext-real ) Real) st r : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) holds
ex g being ( ( ) ( V11() real ext-real ) Real) st
( r : ( ( ) ( V11() real ext-real ) Real) < g : ( ( ) ( V11() real ext-real ) Real) & g : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) ) ) & ( for r being ( ( ) ( V11() real ext-real ) Real) st x0 : ( ( ) ( V11() real ext-real ) Real) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex g being ( ( ) ( V11() real ext-real ) Real) st
( g : ( ( ) ( V11() real ext-real ) Real) < r : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g : ( ( ) ( V11() real ext-real ) Real) & g : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) ) ) ) ) ;

definition
let f be ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let x0 be ( ( ) ( V11() real ext-real ) Real) ;
pred f is_convergent_in x0 means :: LIMFUNC3:def 1
( ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) < x0 : ( ( ) ( ) set ) & x0 : ( ( ) ( ) set ) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( ) set ) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( ) set ) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) : ( ( ) ( ) set ) ) ) & ex g being ( ( ) ( V11() real ext-real ) Real) st
for seq being ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( ) ( V11() real ext-real ) Real) is convergent & lim seq : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) = x0 : ( ( ) ( ) set ) & rng seq : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( ) set ) c= (dom f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) ) : ( ( ) ( ) set ) \ {x0 : ( ( ) ( ) set ) } : ( ( ) ( ) set ) : ( ( ) ( ) Element of K6((dom f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
( f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) /* seq : ( ( ) ( V11() real ext-real ) Real) : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent & lim (f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) /* seq : ( ( ) ( V11() real ext-real ) Real) ) : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) = g : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) );
pred f is_divergent_to+infty_in x0 means :: LIMFUNC3:def 2
( ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) < x0 : ( ( ) ( ) set ) & x0 : ( ( ) ( ) set ) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( ) set ) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( ) set ) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) : ( ( ) ( ) set ) ) ) & ( for seq being ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) = x0 : ( ( ) ( ) set ) & rng seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V69() V70() V71() ) set ) c= (dom f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) ) : ( ( ) ( ) set ) \ {x0 : ( ( ) ( ) set ) } : ( ( ) ( ) set ) : ( ( ) ( ) Element of K6((dom f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) /* seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is divergent_to+infty ) );
pred f is_divergent_to-infty_in x0 means :: LIMFUNC3:def 3
( ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) < x0 : ( ( ) ( ) set ) & x0 : ( ( ) ( ) set ) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( ) set ) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( ) set ) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) : ( ( ) ( ) set ) ) ) & ( for seq being ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) = x0 : ( ( ) ( ) set ) & rng seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V69() V70() V71() ) set ) c= (dom f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) ) : ( ( ) ( ) set ) \ {x0 : ( ( ) ( ) set ) } : ( ( ) ( ) set ) : ( ( ) ( ) Element of K6((dom f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) /* seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is divergent_to-infty ) );
end;

theorem :: LIMFUNC3:9
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) holds
( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) iff ( ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) ) ) & ex g being ( ( ) ( V11() real ext-real ) Real) st
for g1 being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < g1 : ( ( ) ( V11() real ext-real ) Real) holds
ex g2 being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < g2 : ( ( ) ( V11() real ext-real ) Real) & ( for r1 being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < abs (x0 : ( ( ) ( V11() real ext-real ) Real) - r1 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) & abs (x0 : ( ( ) ( V11() real ext-real ) Real) - r1 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) < g2 : ( ( ) ( V11() real ext-real ) Real) & r1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) holds
abs ((f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . r1 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) - g : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) < g1 : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ;

theorem :: LIMFUNC3:10
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) holds
( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) iff ( ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) ) ) & ( for g1 being ( ( ) ( V11() real ext-real ) Real) ex g2 being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < g2 : ( ( ) ( V11() real ext-real ) Real) & ( for r1 being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < abs (x0 : ( ( ) ( V11() real ext-real ) Real) - r1 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) & abs (x0 : ( ( ) ( V11() real ext-real ) Real) - r1 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) < g2 : ( ( ) ( V11() real ext-real ) Real) & r1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) holds
g1 : ( ( ) ( V11() real ext-real ) Real) < f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . r1 : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ) ) ) ) ;

theorem :: LIMFUNC3:11
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) holds
( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) iff ( ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) ) ) & ( for g1 being ( ( ) ( V11() real ext-real ) Real) ex g2 being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < g2 : ( ( ) ( V11() real ext-real ) Real) & ( for r1 being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < abs (x0 : ( ( ) ( V11() real ext-real ) Real) - r1 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) & abs (x0 : ( ( ) ( V11() real ext-real ) Real) - r1 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) < g2 : ( ( ) ( V11() real ext-real ) Real) & r1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) holds
f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . r1 : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) < g1 : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: LIMFUNC3:12
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) holds
( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) iff ( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_left_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_right_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ) ) ;

theorem :: LIMFUNC3:13
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) holds
( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) iff ( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_left_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_right_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ) ) ;

theorem :: LIMFUNC3:14
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in (dom f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (dom f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in (dom f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (dom f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
( f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) + f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ) ;

theorem :: LIMFUNC3:15
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in (dom f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (dom f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in (dom f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (dom f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) holds
( f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) + f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ) ;

theorem :: LIMFUNC3:16
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) + f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) + f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is bounded_below ) holds
f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) + f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: LIMFUNC3:17
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) & ex r, r1 being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r1 : ( ( ) ( V11() real ext-real ) Real) & ( for g being ( ( ) ( V11() real ext-real ) Real) st g : ( ( ) ( V11() real ext-real ) Real) in (dom f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) holds
r1 : ( ( ) ( V11() real ext-real ) Real) <= f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ) holds
f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: LIMFUNC3:18
for x0, r being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) holds
( ( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) implies r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ) & ( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) < 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) implies r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ) & ( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) implies r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ) & ( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) < 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) implies r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ) ) ;

theorem :: LIMFUNC3:19
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st ( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) or f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ) holds
abs f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: LIMFUNC3:20
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st ex r being ( ( ) ( V11() real ext-real ) Real) st
( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is non-decreasing & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is non-increasing & not f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is bounded_above & not f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is bounded_above ) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) ) ) holds
f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: LIMFUNC3:21
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is increasing & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is decreasing & not f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is bounded_above & not f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is bounded_above ) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) ) ) holds
f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: LIMFUNC3:22
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st ex r being ( ( ) ( V11() real ext-real ) Real) st
( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is non-increasing & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is non-decreasing & not f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is bounded_below & not f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is bounded_below ) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) ) ) holds
f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: LIMFUNC3:23
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is decreasing & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is increasing & not f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is bounded_below & not f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is bounded_below ) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) ) ) holds
f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: LIMFUNC3:24
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) ) ) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) c= (dom f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) & ( for g being ( ( ) ( V11() real ext-real ) Real) st g : ( ( ) ( V11() real ext-real ) Real) in (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) holds
f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <= f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ) holds
f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: LIMFUNC3:25
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) ) ) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) c= (dom f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) & ( for g being ( ( ) ( V11() real ext-real ) Real) st g : ( ( ) ( V11() real ext-real ) Real) in (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <= f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ) holds
f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: LIMFUNC3:26
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) c= (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (dom f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) & ( for g being ( ( ) ( V11() real ext-real ) Real) st g : ( ( ) ( V11() real ext-real ) Real) in ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) holds
f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <= f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ) holds
f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: LIMFUNC3:27
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) c= (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (dom f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) & ( for g being ( ( ) ( V11() real ext-real ) Real) st g : ( ( ) ( V11() real ext-real ) Real) in ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <= f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ) holds
f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ;

definition
let f be ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let x0 be ( ( ) ( V11() real ext-real ) Real) ;
assume f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) ;
func lim (f,x0) -> ( ( ) ( V11() real ext-real ) Real) means :: LIMFUNC3:def 4
 for seq being ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) = x0 : ( ( ) ( ) set ) & rng seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V69() V70() V71() ) set ) c= (dom f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) ) : ( ( ) ( ) set ) \ {x0 : ( ( ) ( ) set ) } : ( ( ) ( ) set ) : ( ( ) ( ) Element of K6((dom f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
( f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) /* seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent & lim (f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) /* seq : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) = it : ( ( V21() ) ( Relation-like f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) -defined x0 : ( ( ) ( ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(f : ( ( Relation-like V21() natural-valued ) ( Relation-like RAT : ( ( ) ( non empty V49() V69() V70() V71() V72() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued natural-valued ) set ) ,x0 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) );
end;

theorem :: LIMFUNC3:28
for x0, g being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) holds
( lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = g : ( ( ) ( V11() real ext-real ) Real) iff for g1 being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < g1 : ( ( ) ( V11() real ext-real ) Real) holds
ex g2 being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < g2 : ( ( ) ( V11() real ext-real ) Real) & ( for r1 being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < abs (x0 : ( ( ) ( V11() real ext-real ) Real) - r1 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) & abs (x0 : ( ( ) ( V11() real ext-real ) Real) - r1 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) < g2 : ( ( ) ( V11() real ext-real ) Real) & r1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) holds
abs ((f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . r1 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) - g : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) < g1 : ( ( ) ( V11() real ext-real ) Real) ) ) ) ;

theorem :: LIMFUNC3:29
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) holds
( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_left_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_right_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim_left (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) = lim_right (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) & lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = lim_left (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) & lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = lim_right (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ;

theorem :: LIMFUNC3:30
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_left_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_right_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim_left (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) = lim_right (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) holds
( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = lim_left (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) & lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = lim_right (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ;

theorem :: LIMFUNC3:31
for x0, r being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) holds
( r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim ((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = r : ( ( ) ( V11() real ext-real ) Real) * (lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ;

theorem :: LIMFUNC3:32
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) holds
( - f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim ((- f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = - (lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ;

theorem :: LIMFUNC3:33
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) + f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) + f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) + f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim ((f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) + f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = (lim (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( V11() real ext-real ) Real) + (lim (f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ;

theorem :: LIMFUNC3:34
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) - f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) - f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) - f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim ((f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) - f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = (lim (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( V11() real ext-real ) Real) - (lim (f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ;

theorem :: LIMFUNC3:35
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) " {0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) ( V69() V70() V71() V72() V73() V74() ) set ) : ( ( ) ( ) set ) = {} : ( ( ) ( ) set ) & lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^ : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim ((f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = (lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( V11() real ext-real ) Real) " : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ;

theorem :: LIMFUNC3:36
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) holds
( abs f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim ((abs f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = abs (lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ;

theorem :: LIMFUNC3:37
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g1 : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g2 : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^ : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim ((f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = (lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( V11() real ext-real ) Real) " : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ;

theorem :: LIMFUNC3:38
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim ((f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = (lim (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( V11() real ext-real ) Real) * (lim (f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ;

theorem :: LIMFUNC3:39
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim (f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) / f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) / f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) / f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim ((f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) / f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = (lim (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( V11() real ext-real ) Real) / (lim (f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ;

theorem :: LIMFUNC3:40
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) | (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is bounded ) holds
( f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim ((f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: LIMFUNC3:41
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f2, f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = lim (f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) ) ) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ( for g being ( ( ) ( V11() real ext-real ) Real) st g : ( ( ) ( V11() real ext-real ) Real) in (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) holds
( f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <= f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <= f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ) & ( ( (dom f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) c= (dom f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) & (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) c= (dom f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) or ( (dom f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) c= (dom f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) & (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) c= (dom f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = lim (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) ) ;

theorem :: LIMFUNC3:42
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f2, f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = lim (f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) c= ((dom f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (dom f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) /\ (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) & ( for g being ( ( ) ( V11() real ext-real ) Real) st g : ( ( ) ( V11() real ext-real ) Real) in ].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) holds
( f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <= f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <= f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ) ) holds
( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = lim (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) ) ;

theorem :: LIMFUNC3:43
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f1, f2 being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ( ( (dom f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) c= (dom f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) & ( for g being ( ( ) ( V11() real ext-real ) Real) st g : ( ( ) ( V11() real ext-real ) Real) in (dom f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) holds
f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <= f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ) or ( (dom f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) c= (dom f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) & ( for g being ( ( ) ( V11() real ext-real ) Real) st g : ( ( ) ( V11() real ext-real ) Real) in (dom f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) holds
f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <= f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ) ) ) holds
lim (f1 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) <= lim (f2 : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: LIMFUNC3:44
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st ( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) or f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g1 : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g2 : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^ : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim ((f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^) : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: LIMFUNC3:45
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g1 : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g2 : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ( for g being ( ( ) ( V11() real ext-real ) Real) st g : ( ( ) ( V11() real ext-real ) Real) in (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) holds
0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) <= f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ) holds
f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^ : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: LIMFUNC3:46
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for r1, r2 being ( ( ) ( V11() real ext-real ) Real) st r1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) holds
ex g1, g2 being ( ( ) ( V11() real ext-real ) Real) st
( r1 : ( ( ) ( V11() real ext-real ) Real) < g1 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) < x0 : ( ( ) ( V11() real ext-real ) Real) & g1 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & g2 : ( ( ) ( V11() real ext-real ) Real) < r2 : ( ( ) ( V11() real ext-real ) Real) & x0 : ( ( ) ( V11() real ext-real ) Real) < g2 : ( ( ) ( V11() real ext-real ) Real) & g2 : ( ( ) ( V11() real ext-real ) Real) in dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g1 : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g2 : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ( for g being ( ( ) ( V11() real ext-real ) Real) st g : ( ( ) ( V11() real ext-real ) Real) in (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) <= 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^ : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: LIMFUNC3:47
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ( for g being ( ( ) ( V11() real ext-real ) Real) st g : ( ( ) ( V11() real ext-real ) Real) in (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) holds
0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ) ) holds
f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^ : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_divergent_to+infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: LIMFUNC3:48
for x0 being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) st f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_convergent_in x0 : ( ( ) ( V11() real ext-real ) Real) & lim (f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) & ex r being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & ( for g being ( ( ) ( V11() real ext-real ) Real) st g : ( ( ) ( V11() real ext-real ) Real) in (dom f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( ) set ) /\ (].(x0 : ( ( ) ( V11() real ext-real ) Real) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) ,x0 : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V11() real ext-real ) Real) ,(x0 : ( ( ) ( V11() real ext-real ) Real) + r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) .[ : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V69() V70() V71() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) . g : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) < 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V69() V70() V71() V72() V73() V74() V85() V86() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V69() V70() V71() V72() V73() V74() V75() ) Element of K6(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
f : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^ : ( ( V21() ) ( Relation-like REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -defined REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) -valued V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ,REAL : ( ( ) ( non empty V49() V69() V70() V71() V75() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_divergent_to-infty_in x0 : ( ( ) ( V11() real ext-real ) Real) ;