:: FDIFF_2 semantic presentation
begin
registration
let h be non-zero 0 -convergent Real_Sequence;
cluster - h -> non-zero convergent ;
coherence
( - h is non-empty & - h is convergent )
proof
A1: - h is non-zero by SEQ_1:45;
- h is convergent by SEQ_2:9;
hence ( - h is non-empty & - h is convergent ) by A1; ::_thesis: verum
end;
end;
theorem Th1: :: FDIFF_2:1
for a, b, d being Real_Sequence st a is convergent & b is convergent & lim a = lim b & ( for n being Element of NAT holds
( d . (2 * n) = a . n & d . ((2 * n) + 1) = b . n ) ) holds
( d is convergent & lim d = lim a )
proof
let a, b, d be Real_Sequence; ::_thesis: ( a is convergent & b is convergent & lim a = lim b & ( for n being Element of NAT holds
( d . (2 * n) = a . n & d . ((2 * n) + 1) = b . n ) ) implies ( d is convergent & lim d = lim a ) )
assume that
A1: a is convergent and
A2: b is convergent and
A3: lim a = lim b and
A4: for n being Element of NAT holds
( d . (2 * n) = a . n & d . ((2 * n) + 1) = b . n ) ; ::_thesis: ( d is convergent & lim d = lim a )
A5: now__::_thesis:_for_r_being_real_number_st_0_<_r_holds_
ex_n_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n_<=_m_holds_
abs_((d_._m)_-_(lim_a))_<_r
let r be real number ; ::_thesis: ( 0 < r implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((d . m) - (lim a)) < r )
assume A6: 0 < r ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((d . m) - (lim a)) < r
then consider k1 being Element of NAT such that
A7: for m being Element of NAT st k1 <= m holds
abs ((a . m) - (lim a)) < r by A1, SEQ_2:def_7;
consider k2 being Element of NAT such that
A8: for m being Element of NAT st k2 <= m holds
abs ((b . m) - (lim b)) < r by A2, A6, SEQ_2:def_7;
take n = max ((2 * k1),((2 * k2) + 1)); ::_thesis: for m being Element of NAT st n <= m holds
abs ((d . m) - (lim a)) < r
let m be Element of NAT ; ::_thesis: ( n <= m implies abs ((d . m) - (lim a)) < r )
assume A9: n <= m ; ::_thesis: abs ((d . m) - (lim a)) < r
then A10: (2 * k2) + 1 <= m by XXREAL_0:30;
consider n being Element of NAT such that
A11: ( m = 2 * n or m = (2 * n) + 1 ) by SCHEME1:1;
A12: 2 * k1 <= m by A9, XXREAL_0:30;
now__::_thesis:_abs_((d_._m)_-_(lim_a))_<_r
percases ( m = 2 * n or m = (2 * n) + 1 ) by A11;
supposeA13: m = 2 * n ; ::_thesis: abs ((d . m) - (lim a)) < r
then A14: n >= k1 by A12, XREAL_1:68;
abs ((d . m) - (lim a)) = abs ((a . n) - (lim a)) by A4, A13;
hence abs ((d . m) - (lim a)) < r by A7, A14; ::_thesis: verum
end;
supposeA15: m = (2 * n) + 1 ; ::_thesis: abs ((d . m) - (lim a)) < r
A16: now__::_thesis:_not_n_<_k2
assume n < k2 ; ::_thesis: contradiction
then 2 * n < 2 * k2 by XREAL_1:68;
hence contradiction by A10, A15, XREAL_1:6; ::_thesis: verum
end;
abs ((d . m) - (lim a)) = abs ((b . n) - (lim a)) by A4, A15;
hence abs ((d . m) - (lim a)) < r by A3, A8, A16; ::_thesis: verum
end;
end;
end;
hence abs ((d . m) - (lim a)) < r ; ::_thesis: verum
end;
hence d is convergent by SEQ_2:def_6; ::_thesis: lim d = lim a
hence lim d = lim a by A5, SEQ_2:def_7; ::_thesis: verum
end;
theorem Th2: :: FDIFF_2:2
for a being Real_Sequence st ( for n being Element of NAT holds a . n = 2 * n ) holds
a is V37() sequence of NAT
proof
let a be Real_Sequence; ::_thesis: ( ( for n being Element of NAT holds a . n = 2 * n ) implies a is V37() sequence of NAT )
assume A1: for n being Element of NAT holds a . n = 2 * n ; ::_thesis: a is V37() sequence of NAT
A2: a is V37()
proof
let n be Element of NAT ; :: according to SEQM_3:def_6 ::_thesis: not a . (n + 1) <= a . n
A3: (2 * n) + 0 < (2 * n) + 2 by XREAL_1:8;
(2 * n) + 2 = 2 * (n + 1)
.= a . (n + 1) by A1 ;
hence a . n < a . (n + 1) by A1, A3; ::_thesis: verum
end;
A4: now__::_thesis:_for_x_being_set_st_x_in_NAT_holds_
a_._x_in_NAT
let x be set ; ::_thesis: ( x in NAT implies a . x in NAT )
assume x in NAT ; ::_thesis: a . x in NAT
then reconsider n = x as Element of NAT ;
a . n = 2 * n by A1;
hence a . x in NAT ; ::_thesis: verum
end;
dom a = NAT by FUNCT_2:def_1;
hence a is V37() sequence of NAT by A2, A4, FUNCT_2:3; ::_thesis: verum
end;
theorem Th3: :: FDIFF_2:3
for a being Real_Sequence st ( for n being Element of NAT holds a . n = (2 * n) + 1 ) holds
a is V37() sequence of NAT
proof
let a be Real_Sequence; ::_thesis: ( ( for n being Element of NAT holds a . n = (2 * n) + 1 ) implies a is V37() sequence of NAT )
assume A1: for n being Element of NAT holds a . n = (2 * n) + 1 ; ::_thesis: a is V37() sequence of NAT
A2: a is V37()
proof
let n be Element of NAT ; :: according to SEQM_3:def_6 ::_thesis: not a . (n + 1) <= a . n
A3: ((2 * n) + 1) + 0 < ((2 * n) + 1) + 2 by XREAL_1:8;
((2 * n) + 1) + 2 = (2 * (n + 1)) + 1
.= a . (n + 1) by A1 ;
hence a . n < a . (n + 1) by A1, A3; ::_thesis: verum
end;
A4: now__::_thesis:_for_x_being_set_st_x_in_dom_a_holds_
a_._x_in_NAT
let x be set ; ::_thesis: ( x in dom a implies a . x in NAT )
assume x in dom a ; ::_thesis: a . x in NAT
then reconsider n = x as Element of NAT ;
a . n = (2 * n) + 1 by A1;
hence a . x in NAT ; ::_thesis: verum
end;
dom a = NAT by FUNCT_2:def_1;
hence a is V37() sequence of NAT by A2, A4, FUNCT_2:3; ::_thesis: verum
end;
theorem Th4: :: FDIFF_2:4
for x0 being Real
for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} holds
( c is convergent & lim c = x0 & h + c is convergent & lim (h + c) = x0 )
proof
let x0 be Real; ::_thesis: for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} holds
( c is convergent & lim c = x0 & h + c is convergent & lim (h + c) = x0 )
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: for c being V8() Real_Sequence st rng c = {x0} holds
( c is convergent & lim c = x0 & h + c is convergent & lim (h + c) = x0 )
let c be V8() Real_Sequence; ::_thesis: ( rng c = {x0} implies ( c is convergent & lim c = x0 & h + c is convergent & lim (h + c) = x0 ) )
assume A1: rng c = {x0} ; ::_thesis: ( c is convergent & lim c = x0 & h + c is convergent & lim (h + c) = x0 )
thus c is convergent ; ::_thesis: ( lim c = x0 & h + c is convergent & lim (h + c) = x0 )
x0 in rng c by A1, TARSKI:def_1;
hence A2: lim c = x0 by SEQ_4:25; ::_thesis: ( h + c is convergent & lim (h + c) = x0 )
thus h + c is convergent by SEQ_2:5; ::_thesis: lim (h + c) = x0
lim h = 0 ;
hence lim (h + c) = 0 + x0 by A2, SEQ_2:6
.= x0 ;
::_thesis: verum
end;
theorem Th5: :: FDIFF_2:5
for r being Real
for a, b being Real_Sequence st rng a = {r} & rng b = {r} holds
a = b
proof
let r be Real; ::_thesis: for a, b being Real_Sequence st rng a = {r} & rng b = {r} holds
a = b
let a, b be Real_Sequence; ::_thesis: ( rng a = {r} & rng b = {r} implies a = b )
assume that
A1: rng a = {r} and
A2: rng b = {r} ; ::_thesis: a = b
now__::_thesis:_for_n_being_Element_of_NAT_holds_a_._n_=_b_._n
let n be Element of NAT ; ::_thesis: a . n = b . n
a . n in rng a by VALUED_0:28;
then A3: a . n = r by A1, TARSKI:def_1;
b . n in rng b by VALUED_0:28;
hence a . n = b . n by A2, A3, TARSKI:def_1; ::_thesis: verum
end;
hence a = b by FUNCT_2:63; ::_thesis: verum
end;
theorem Th6: :: FDIFF_2:6
for a being Real_Sequence
for h being non-zero 0 -convergent Real_Sequence st a is subsequence of h holds
a is non-zero 0 -convergent Real_Sequence
proof
let a be Real_Sequence; ::_thesis: for h being non-zero 0 -convergent Real_Sequence st a is subsequence of h holds
a is non-zero 0 -convergent Real_Sequence
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: ( a is subsequence of h implies a is non-zero 0 -convergent Real_Sequence )
assume A1: a is subsequence of h ; ::_thesis: a is non-zero 0 -convergent Real_Sequence
then consider I being V37() sequence of NAT such that
A2: a = h * I by VALUED_0:def_17;
now__::_thesis:_for_n_being_Element_of_NAT_holds_not_a_._n_=_0
given n being Element of NAT such that A4: a . n = 0 ; ::_thesis: contradiction
h . (I . n) <> 0 by SEQ_1:5;
hence contradiction by A2, A4, FUNCT_2:15; ::_thesis: verum
end;
then A5: a is non-zero by SEQ_1:5;
A7: a is convergent by A1, SEQ_4:16;
lim h = 0 ;
then lim a = 0 by A1, SEQ_4:17;
hence a is non-zero 0 -convergent Real_Sequence by A7, A5, FDIFF_1:def_1; ::_thesis: verum
end;
theorem Th7: :: FDIFF_2:7
for g being Real
for f being PartFunc of REAL,REAL st ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {g} & rng (h + c) c= dom f & {g} c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ) holds
for h1, h2 being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {g} & rng (h1 + c) c= dom f & rng (h2 + c) c= dom f & {g} c= dom f holds
lim ((h1 ") (#) ((f /* (h1 + c)) - (f /* c))) = lim ((h2 ") (#) ((f /* (h2 + c)) - (f /* c)))
proof
let g be Real; ::_thesis: for f being PartFunc of REAL,REAL st ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {g} & rng (h + c) c= dom f & {g} c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ) holds
for h1, h2 being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {g} & rng (h1 + c) c= dom f & rng (h2 + c) c= dom f & {g} c= dom f holds
lim ((h1 ") (#) ((f /* (h1 + c)) - (f /* c))) = lim ((h2 ") (#) ((f /* (h2 + c)) - (f /* c)))
let f be PartFunc of REAL,REAL; ::_thesis: ( ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {g} & rng (h + c) c= dom f & {g} c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ) implies for h1, h2 being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {g} & rng (h1 + c) c= dom f & rng (h2 + c) c= dom f & {g} c= dom f holds
lim ((h1 ") (#) ((f /* (h1 + c)) - (f /* c))) = lim ((h2 ") (#) ((f /* (h2 + c)) - (f /* c))) )
assume A1: for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {g} & rng (h + c) c= dom f & {g} c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ; ::_thesis: for h1, h2 being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {g} & rng (h1 + c) c= dom f & rng (h2 + c) c= dom f & {g} c= dom f holds
lim ((h1 ") (#) ((f /* (h1 + c)) - (f /* c))) = lim ((h2 ") (#) ((f /* (h2 + c)) - (f /* c)))
let h1, h2 be non-zero 0 -convergent Real_Sequence; ::_thesis: for c being V8() Real_Sequence st rng c = {g} & rng (h1 + c) c= dom f & rng (h2 + c) c= dom f & {g} c= dom f holds
lim ((h1 ") (#) ((f /* (h1 + c)) - (f /* c))) = lim ((h2 ") (#) ((f /* (h2 + c)) - (f /* c)))
let c be V8() Real_Sequence; ::_thesis: ( rng c = {g} & rng (h1 + c) c= dom f & rng (h2 + c) c= dom f & {g} c= dom f implies lim ((h1 ") (#) ((f /* (h1 + c)) - (f /* c))) = lim ((h2 ") (#) ((f /* (h2 + c)) - (f /* c))) )
assume that
A2: rng c = {g} and
A3: rng (h1 + c) c= dom f and
A4: rng (h2 + c) c= dom f and
A5: {g} c= dom f ; ::_thesis: lim ((h1 ") (#) ((f /* (h1 + c)) - (f /* c))) = lim ((h2 ") (#) ((f /* (h2 + c)) - (f /* c)))
deffunc H1( Element of NAT ) -> Element of REAL = h2 . $1;
deffunc H2( Element of NAT ) -> Element of REAL = h1 . $1;
consider a being Real_Sequence such that
A6: for n being Element of NAT holds
( a . (2 * n) = H2(n) & a . ((2 * n) + 1) = H1(n) ) from SCHEME1:sch_2();
now__::_thesis:_for_n_being_Element_of_NAT_holds_a_._n_<>_0
let n be Element of NAT ; ::_thesis: a . n <> 0
consider m being Element of NAT such that
A9: ( n = 2 * m or n = (2 * m) + 1 ) by SCHEME1:1;
now__::_thesis:_a_._n_<>_0
percases ( n = 2 * m or n = (2 * m) + 1 ) by A9;
suppose n = 2 * m ; ::_thesis: a . n <> 0
then a . n = h1 . m by A6;
hence a . n <> 0 by SEQ_1:5; ::_thesis: verum
end;
suppose n = (2 * m) + 1 ; ::_thesis: a . n <> 0
then a . n = h2 . m by A6;
hence a . n <> 0 by SEQ_1:5; ::_thesis: verum
end;
end;
end;
hence a . n <> 0 ; ::_thesis: verum
end;
then A10: a is non-zero by SEQ_1:5;
A11: lim h1 = 0 ;
A12: lim h2 = 0 ;
A15: a is convergent by A6, A11, A12, Th1;
lim a = 0 by A6, A11, A12, Th1;
then reconsider a = a as non-zero 0 -convergent Real_Sequence by A15, A10, FDIFF_1:def_1;
A16: rng (a + c) c= dom f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (a + c) or x in dom f )
assume x in rng (a + c) ; ::_thesis: x in dom f
then consider n being Element of NAT such that
A17: x = (a + c) . n by FUNCT_2:113;
consider m being Element of NAT such that
A18: ( n = 2 * m or n = (2 * m) + 1 ) by SCHEME1:1;
now__::_thesis:_(a_+_c)_._n_in_dom_f
percases ( n = 2 * m or n = (2 * m) + 1 ) by A18;
supposeA19: n = 2 * m ; ::_thesis: (a + c) . n in dom f
A20: (h1 + c) . m in rng (h1 + c) by VALUED_0:28;
(a + c) . n = (a . n) + (c . n) by SEQ_1:7
.= (h1 . m) + (c . n) by A6, A19
.= (h1 . m) + (c . m) by VALUED_0:23
.= (h1 + c) . m by SEQ_1:7 ;
hence (a + c) . n in dom f by A3, A20; ::_thesis: verum
end;
supposeA21: n = (2 * m) + 1 ; ::_thesis: (a + c) . n in dom f
A22: (h2 + c) . m in rng (h2 + c) by VALUED_0:28;
(a + c) . n = (a . n) + (c . n) by SEQ_1:7
.= (h2 . m) + (c . n) by A6, A21
.= (h2 . m) + (c . m) by VALUED_0:23
.= (h2 + c) . m by SEQ_1:7 ;
hence (a + c) . n in dom f by A4, A22; ::_thesis: verum
end;
end;
end;
hence x in dom f by A17; ::_thesis: verum
end;
then A23: (a ") (#) ((f /* (a + c)) - (f /* c)) is convergent by A1, A2, A5;
deffunc H3( Element of NAT ) -> Element of NAT = (2 * $1) + 1;
consider d being Real_Sequence such that
A24: for n being Element of NAT holds d . n = H3(n) from SEQ_1:sch_1();
reconsider I2 = d as V37() sequence of NAT by A24, Th3;
now__::_thesis:_for_n_being_Element_of_NAT_holds_(((a_")_(#)_((f_/*_(a_+_c))_-_(f_/*_c)))_*_I2)_._n_=_((h2_")_(#)_((f_/*_(h2_+_c))_-_(f_/*_c)))_._n
let n be Element of NAT ; ::_thesis: (((a ") (#) ((f /* (a + c)) - (f /* c))) * I2) . n = ((h2 ") (#) ((f /* (h2 + c)) - (f /* c))) . n
thus (((a ") (#) ((f /* (a + c)) - (f /* c))) * I2) . n = ((a ") (#) ((f /* (a + c)) - (f /* c))) . (I2 . n) by FUNCT_2:15
.= ((a ") (#) ((f /* (a + c)) - (f /* c))) . ((2 * n) + 1) by A24
.= ((a ") . ((2 * n) + 1)) * (((f /* (a + c)) - (f /* c)) . ((2 * n) + 1)) by SEQ_1:8
.= ((a ") . ((2 * n) + 1)) * (((f /* (a + c)) . ((2 * n) + 1)) - ((f /* c) . ((2 * n) + 1))) by RFUNCT_2:1
.= ((a ") . ((2 * n) + 1)) * ((f . ((a + c) . ((2 * n) + 1))) - ((f /* c) . ((2 * n) + 1))) by A16, FUNCT_2:108
.= ((a ") . ((2 * n) + 1)) * ((f . ((a . ((2 * n) + 1)) + (c . ((2 * n) + 1)))) - ((f /* c) . ((2 * n) + 1))) by SEQ_1:7
.= ((a ") . ((2 * n) + 1)) * ((f . ((h2 . n) + (c . ((2 * n) + 1)))) - ((f /* c) . ((2 * n) + 1))) by A6
.= ((a ") . ((2 * n) + 1)) * ((f . ((h2 . n) + (c . n))) - ((f /* c) . ((2 * n) + 1))) by VALUED_0:23
.= ((a ") . ((2 * n) + 1)) * ((f . ((h2 + c) . n)) - ((f /* c) . ((2 * n) + 1))) by SEQ_1:7
.= ((a ") . ((2 * n) + 1)) * (((f /* (h2 + c)) . n) - ((f /* c) . ((2 * n) + 1))) by A4, FUNCT_2:108
.= ((a . ((2 * n) + 1)) ") * (((f /* (h2 + c)) . n) - ((f /* c) . ((2 * n) + 1))) by VALUED_1:10
.= ((h2 . n) ") * (((f /* (h2 + c)) . n) - ((f /* c) . ((2 * n) + 1))) by A6
.= ((h2 ") . n) * (((f /* (h2 + c)) . n) - ((f /* c) . ((2 * n) + 1))) by VALUED_1:10
.= ((h2 ") . n) * (((f /* (h2 + c)) . n) - (f . (c . ((2 * n) + 1)))) by A2, A5, FUNCT_2:108
.= ((h2 ") . n) * (((f /* (h2 + c)) . n) - (f . (c . n))) by VALUED_0:23
.= ((h2 ") . n) * (((f /* (h2 + c)) . n) - ((f /* c) . n)) by A2, A5, FUNCT_2:108
.= ((h2 ") . n) * (((f /* (h2 + c)) - (f /* c)) . n) by RFUNCT_2:1
.= ((h2 ") (#) ((f /* (h2 + c)) - (f /* c))) . n by SEQ_1:8 ; ::_thesis: verum
end;
then A25: ((a ") (#) ((f /* (a + c)) - (f /* c))) * I2 = (h2 ") (#) ((f /* (h2 + c)) - (f /* c)) by FUNCT_2:63;
((a ") (#) ((f /* (a + c)) - (f /* c))) * I2 is subsequence of (a ") (#) ((f /* (a + c)) - (f /* c)) by VALUED_0:def_17;
then A26: lim (((a ") (#) ((f /* (a + c)) - (f /* c))) * I2) = lim ((a ") (#) ((f /* (a + c)) - (f /* c))) by A23, SEQ_4:17;
deffunc H4( Element of NAT ) -> Element of NAT = 2 * $1;
consider b being Real_Sequence such that
A27: for n being Element of NAT holds b . n = H4(n) from SEQ_1:sch_1();
reconsider I1 = b as V37() sequence of NAT by A27, Th2;
now__::_thesis:_for_n_being_Element_of_NAT_holds_(((a_")_(#)_((f_/*_(a_+_c))_-_(f_/*_c)))_*_I1)_._n_=_((h1_")_(#)_((f_/*_(h1_+_c))_-_(f_/*_c)))_._n
let n be Element of NAT ; ::_thesis: (((a ") (#) ((f /* (a + c)) - (f /* c))) * I1) . n = ((h1 ") (#) ((f /* (h1 + c)) - (f /* c))) . n
thus (((a ") (#) ((f /* (a + c)) - (f /* c))) * I1) . n = ((a ") (#) ((f /* (a + c)) - (f /* c))) . (I1 . n) by FUNCT_2:15
.= ((a ") (#) ((f /* (a + c)) - (f /* c))) . (2 * n) by A27
.= ((a ") . (2 * n)) * (((f /* (a + c)) - (f /* c)) . (2 * n)) by SEQ_1:8
.= ((a ") . (2 * n)) * (((f /* (a + c)) . (2 * n)) - ((f /* c) . (2 * n))) by RFUNCT_2:1
.= ((a ") . (2 * n)) * ((f . ((a + c) . (2 * n))) - ((f /* c) . (2 * n))) by A16, FUNCT_2:108
.= ((a ") . (2 * n)) * ((f . ((a . (2 * n)) + (c . (2 * n)))) - ((f /* c) . (2 * n))) by SEQ_1:7
.= ((a ") . (2 * n)) * ((f . ((h1 . n) + (c . (2 * n)))) - ((f /* c) . (2 * n))) by A6
.= ((a ") . (2 * n)) * ((f . ((h1 . n) + (c . n))) - ((f /* c) . (2 * n))) by VALUED_0:23
.= ((a ") . (2 * n)) * ((f . ((h1 + c) . n)) - ((f /* c) . (2 * n))) by SEQ_1:7
.= ((a ") . (2 * n)) * (((f /* (h1 + c)) . n) - ((f /* c) . (2 * n))) by A3, FUNCT_2:108
.= ((a . (2 * n)) ") * (((f /* (h1 + c)) . n) - ((f /* c) . (2 * n))) by VALUED_1:10
.= ((h1 . n) ") * (((f /* (h1 + c)) . n) - ((f /* c) . (2 * n))) by A6
.= ((h1 ") . n) * (((f /* (h1 + c)) . n) - ((f /* c) . (2 * n))) by VALUED_1:10
.= ((h1 ") . n) * (((f /* (h1 + c)) . n) - (f . (c . (2 * n)))) by A2, A5, FUNCT_2:108
.= ((h1 ") . n) * (((f /* (h1 + c)) . n) - (f . (c . n))) by VALUED_0:23
.= ((h1 ") . n) * (((f /* (h1 + c)) . n) - ((f /* c) . n)) by A2, A5, FUNCT_2:108
.= ((h1 ") . n) * (((f /* (h1 + c)) - (f /* c)) . n) by RFUNCT_2:1
.= ((h1 ") (#) ((f /* (h1 + c)) - (f /* c))) . n by SEQ_1:8 ; ::_thesis: verum
end;
then A28: ((a ") (#) ((f /* (a + c)) - (f /* c))) * I1 = (h1 ") (#) ((f /* (h1 + c)) - (f /* c)) by FUNCT_2:63;
((a ") (#) ((f /* (a + c)) - (f /* c))) * I1 is subsequence of (a ") (#) ((f /* (a + c)) - (f /* c)) by VALUED_0:def_17;
hence lim ((h1 ") (#) ((f /* (h1 + c)) - (f /* c))) = lim ((h2 ") (#) ((f /* (h2 + c)) - (f /* c))) by A23, A26, A28, A25, SEQ_4:17; ::_thesis: verum
end;
theorem Th8: :: FDIFF_2:8
for r being Real
for f being PartFunc of REAL,REAL st ex N being Neighbourhood of r st N c= dom f holds
ex h being non-zero 0 -convergent Real_Sequence ex c being V8() Real_Sequence st
( rng c = {r} & rng (h + c) c= dom f & {r} c= dom f )
proof
let r be Real; ::_thesis: for f being PartFunc of REAL,REAL st ex N being Neighbourhood of r st N c= dom f holds
ex h being non-zero 0 -convergent Real_Sequence ex c being V8() Real_Sequence st
( rng c = {r} & rng (h + c) c= dom f & {r} c= dom f )
let f be PartFunc of REAL,REAL; ::_thesis: ( ex N being Neighbourhood of r st N c= dom f implies ex h being non-zero 0 -convergent Real_Sequence ex c being V8() Real_Sequence st
( rng c = {r} & rng (h + c) c= dom f & {r} c= dom f ) )
given N being Neighbourhood of r such that A1: N c= dom f ; ::_thesis: ex h being non-zero 0 -convergent Real_Sequence ex c being V8() Real_Sequence st
( rng c = {r} & rng (h + c) c= dom f & {r} c= dom f )
reconsider a = NAT --> r as Real_Sequence by FUNCOP_1:45;
consider g being real number such that
A2: 0 < g and
A3: N = ].(r - g),(r + g).[ by RCOMP_1:def_6;
reconsider a = a as V8() Real_Sequence ;
deffunc H1( Element of NAT ) -> Element of REAL = g / ($1 + 2);
consider b being Real_Sequence such that
A4: for n being Element of NAT holds b . n = H1(n) from SEQ_1:sch_1();
A5: lim b = 0 by A4, SEQ_4:31;
A6: b is convergent by A4, SEQ_4:31;
now__::_thesis:_for_n_being_Element_of_NAT_holds_0_<>_b_._n
let n be Element of NAT ; ::_thesis: 0 <> b . n
0 < g / (n + 2) by A2;
hence 0 <> b . n by A4; ::_thesis: verum
end;
then b is non-zero by SEQ_1:5;
then reconsider b = b as non-zero 0 -convergent Real_Sequence by A6, A5, FDIFF_1:def_1;
take b ; ::_thesis: ex c being V8() Real_Sequence st
( rng c = {r} & rng (b + c) c= dom f & {r} c= dom f )
take a ; ::_thesis: ( rng a = {r} & rng (b + a) c= dom f & {r} c= dom f )
thus rng a = {r} ::_thesis: ( rng (b + a) c= dom f & {r} c= dom f )
proof
thus rng a c= {r} :: according to XBOOLE_0:def_10 ::_thesis: {r} c= rng a
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng a or x in {r} )
assume x in rng a ; ::_thesis: x in {r}
then ex n being Element of NAT st x = a . n by FUNCT_2:113;
then x = r by FUNCOP_1:7;
hence x in {r} by TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {r} or x in rng a )
assume x in {r} ; ::_thesis: x in rng a
then x = r by TARSKI:def_1
.= a . 0 by FUNCOP_1:7 ;
hence x in rng a by VALUED_0:28; ::_thesis: verum
end;
thus rng (b + a) c= dom f ::_thesis: {r} c= dom f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (b + a) or x in dom f )
assume x in rng (b + a) ; ::_thesis: x in dom f
then consider n being Element of NAT such that
A7: x = (b + a) . n by FUNCT_2:113;
0 + 1 < n + 2 by XREAL_1:8;
then g * 1 < g * (n + 2) by A2, XREAL_1:97;
then g * ((n + 2) ") < (g * (n + 2)) * ((n + 2) ") by XREAL_1:68;
then g * ((n + 2) ") < g * ((n + 2) * ((n + 2) ")) ;
then g * ((n + 2) ") < g * 1 by XCMPLX_0:def_7;
then g / (n + 2) < g by XCMPLX_0:def_9;
then A8: r + (g / (n + 2)) < r + g by XREAL_1:6;
A9: r - g < r - 0 by A2, XREAL_1:15;
r + 0 < r + (g / (n + 2)) by A2, XREAL_1:8;
then r - g < r + (g / (n + 2)) by A9, XXREAL_0:2;
then A10: r + (g / (n + 2)) in { g1 where g1 is Real : ( r - g < g1 & g1 < r + g ) } by A8;
x = (b . n) + (a . n) by A7, SEQ_1:7
.= (g / (n + 2)) + (a . n) by A4
.= (g / (n + 2)) + r by FUNCOP_1:7 ;
then x in N by A3, A10, RCOMP_1:def_2;
hence x in dom f by A1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {r} or x in dom f )
assume x in {r} ; ::_thesis: x in dom f
then x = r by TARSKI:def_1;
then x in N by RCOMP_1:16;
hence x in dom f by A1; ::_thesis: verum
end;
theorem Th9: :: FDIFF_2:9
for a being Real_Sequence
for f2, f1 being PartFunc of REAL,REAL st rng a c= dom (f2 * f1) holds
( rng a c= dom f1 & rng (f1 /* a) c= dom f2 )
proof
let a be Real_Sequence; ::_thesis: for f2, f1 being PartFunc of REAL,REAL st rng a c= dom (f2 * f1) holds
( rng a c= dom f1 & rng (f1 /* a) c= dom f2 )
let f2, f1 be PartFunc of REAL,REAL; ::_thesis: ( rng a c= dom (f2 * f1) implies ( rng a c= dom f1 & rng (f1 /* a) c= dom f2 ) )
assume A1: rng a c= dom (f2 * f1) ; ::_thesis: ( rng a c= dom f1 & rng (f1 /* a) c= dom f2 )
then A2: f1 .: (rng a) c= dom f2 by FUNCT_1:101;
rng a c= dom f1 by A1, FUNCT_1:101;
hence ( rng a c= dom f1 & rng (f1 /* a) c= dom f2 ) by A2, VALUED_0:30; ::_thesis: verum
end;
scheme :: FDIFF_2:sch 1
ExIncSeqofNat{ F1() -> Real_Sequence, P1[ set ] } :
ex q being V37() sequence of NAT st
( ( for n being Element of NAT holds P1[(F1() * q) . n] ) & ( for n being Element of NAT st ( for r being Real st r = F1() . n holds
P1[r] ) holds
ex m being Element of NAT st n = q . m ) )
provided
A1: for n being Element of NAT ex m being Element of NAT st
( n <= m & P1[F1() . m] )
proof
defpred S1[ Nat, set , set ] means for n, m being Element of NAT st $2 = n & $3 = m holds
( n < m & P1[F1() . m] & ( for k being Element of NAT st n < k & P1[F1() . k] holds
m <= k ) );
defpred S2[ Nat] means P1[F1() . $1];
ex m1 being Element of NAT st
( 0 <= m1 & P1[F1() . m1] ) by A1;
then A2: ex m being Nat st S2[m] ;
consider M being Nat such that
A3: ( S2[M] & ( for n being Nat st S2[n] holds
M <= n ) ) from NAT_1:sch_5(A2);
reconsider M9 = M as Element of NAT by ORDINAL1:def_12;
A4: now__::_thesis:_for_n_being_Element_of_NAT_ex_m_being_Element_of_NAT_st_
(_n_<_m_&_P1[F1()_._m]_)
let n be Element of NAT ; ::_thesis: ex m being Element of NAT st
( n < m & P1[F1() . m] )
consider m being Element of NAT such that
A5: n + 1 <= m and
A6: P1[F1() . m] by A1;
take m = m; ::_thesis: ( n < m & P1[F1() . m] )
thus ( n < m & P1[F1() . m] ) by A5, A6, NAT_1:13; ::_thesis: verum
end;
A7: for n, x being Element of NAT ex y being Element of NAT st S1[n,x,y]
proof
let n, x be Element of NAT ; ::_thesis: ex y being Element of NAT st S1[n,x,y]
defpred S3[ Nat] means ( x < $1 & P1[F1() . $1] );
ex m being Element of NAT st S3[m] by A4;
then A8: ex m being Nat st S3[m] ;
consider l being Nat such that
A9: ( S3[l] & ( for k being Nat st S3[k] holds
l <= k ) ) from NAT_1:sch_5(A8);
take l ; ::_thesis: ( l is Element of REAL & l is Element of NAT & S1[n,x,l] )
l in NAT by ORDINAL1:def_12;
hence ( l is Element of REAL & l is Element of NAT & S1[n,x,l] ) by A9; ::_thesis: verum
end;
consider F being Function of NAT,NAT such that
A10: ( F . 0 = M9 & ( for n being Element of NAT holds S1[n,F . n,F . (n + 1)] ) ) from RECDEF_1:sch_2(A7);
A11: rng F c= REAL by XBOOLE_1:1;
A12: rng F c= NAT ;
A13: dom F = NAT by FUNCT_2:def_1;
then reconsider F = F as Real_Sequence by A11, RELSET_1:4;
A14: now__::_thesis:_for_n_being_Element_of_NAT_holds_F_._n_is_Element_of_NAT
let n be Element of NAT ; ::_thesis: F . n is Element of NAT
F . n in rng F by A13, FUNCT_1:def_3;
hence F . n is Element of NAT by A12; ::_thesis: verum
end;
now__::_thesis:_for_n_being_Element_of_NAT_holds_F_._n_<_F_._(n_+_1)
let n be Element of NAT ; ::_thesis: F . n < F . (n + 1)
A15: F . (n + 1) is Element of NAT by A14;
F . n is Element of NAT by A14;
hence F . n < F . (n + 1) by A10, A15; ::_thesis: verum
end;
then reconsider F = F as V37() sequence of NAT by SEQM_3:def_6;
A16: for n being Element of NAT st P1[F1() . n] holds
ex m being Element of NAT st F . m = n
proof
defpred S3[ Nat] means ( P1[F1() . $1] & ( for m being Element of NAT holds F . m <> $1 ) );
assume ex n being Element of NAT st S3[n] ; ::_thesis: contradiction
then A17: ex n being Nat st S3[n] ;
consider M1 being Nat such that
A18: ( S3[M1] & ( for n being Nat st S3[n] holds
M1 <= n ) ) from NAT_1:sch_5(A17);
defpred S4[ Nat] means ( $1 < M1 & P1[F1() . $1] & ex m being Element of NAT st F . m = $1 );
A19: ex n being Nat st S4[n]
proof
take M ; ::_thesis: S4[M]
A20: M <> M1 by A10, A18;
M <= M1 by A3, A18;
hence M < M1 by A20, XXREAL_0:1; ::_thesis: ( P1[F1() . M] & ex m being Element of NAT st F . m = M )
thus P1[F1() . M] by A3; ::_thesis: ex m being Element of NAT st F . m = M
take 0 ; ::_thesis: F . 0 = M
thus F . 0 = M by A10; ::_thesis: verum
end;
A21: for n being Nat st S4[n] holds
n <= M1 ;
consider X being Nat such that
A22: ( S4[X] & ( for n being Nat st S4[n] holds
n <= X ) ) from NAT_1:sch_6(A21, A19);
A23: for k being Element of NAT st X < k & k < M1 holds
not P1[F1() . k]
proof
given k being Element of NAT such that A24: X < k and
A25: k < M1 and
A26: P1[F1() . k] ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ex m being Element of NAT st F . m = k or for m being Element of NAT holds F . m <> k ) ;
suppose ex m being Element of NAT st F . m = k ; ::_thesis: contradiction
hence contradiction by A22, A24, A25, A26; ::_thesis: verum
end;
suppose for m being Element of NAT holds F . m <> k ; ::_thesis: contradiction
hence contradiction by A18, A25, A26; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
consider m being Element of NAT such that
A27: F . m = X by A22;
M1 in NAT by ORDINAL1:def_12;
then A28: F . (m + 1) <= M1 by A10, A18, A22, A27;
A29: P1[F1() . (F . (m + 1))] by A10, A27;
A30: X < F . (m + 1) by A10, A27;
now__::_thesis:_not_F_._(m_+_1)_<>_M1
assume F . (m + 1) <> M1 ; ::_thesis: contradiction
then F . (m + 1) < M1 by A28, XXREAL_0:1;
hence contradiction by A23, A30, A29; ::_thesis: verum
end;
hence contradiction by A18; ::_thesis: verum
end;
take F ; ::_thesis: ( ( for n being Element of NAT holds P1[(F1() * F) . n] ) & ( for n being Element of NAT st ( for r being Real st r = F1() . n holds
P1[r] ) holds
ex m being Element of NAT st n = F . m ) )
set q = F1() * F;
defpred S3[ Nat] means P1[(F1() * F) . $1];
A31: for k being Element of NAT st S3[k] holds
S3[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S3[k] implies S3[k + 1] )
assume P1[(F1() * F) . k] ; ::_thesis: S3[k + 1]
S1[k,F . k,F . (k + 1)] by A10;
then P1[F1() . (F . (k + 1))] ;
hence S3[k + 1] by FUNCT_2:15; ::_thesis: verum
end;
A32: S3[ 0 ] by A3, A10, FUNCT_2:15;
thus for n being Element of NAT holds S3[n] from NAT_1:sch_1(A32, A31); ::_thesis: for n being Element of NAT st ( for r being Real st r = F1() . n holds
P1[r] ) holds
ex m being Element of NAT st n = F . m
let n be Element of NAT ; ::_thesis: ( ( for r being Real st r = F1() . n holds
P1[r] ) implies ex m being Element of NAT st n = F . m )
assume for r being Real st r = F1() . n holds
P1[r] ; ::_thesis: ex m being Element of NAT st n = F . m
then consider m being Element of NAT such that
A33: F . m = n by A16;
take m ; ::_thesis: n = F . m
thus n = F . m by A33; ::_thesis: verum
end;
theorem :: FDIFF_2:10
for x0, r being Real
for f being PartFunc of REAL,REAL st f . x0 <> r & f is_differentiable_in x0 holds
ex N being Neighbourhood of x0 st
( N c= dom f & ( for g being Real st g in N holds
f . g <> r ) )
proof
let x0, r be Real; ::_thesis: for f being PartFunc of REAL,REAL st f . x0 <> r & f is_differentiable_in x0 holds
ex N being Neighbourhood of x0 st
( N c= dom f & ( for g being Real st g in N holds
f . g <> r ) )
let f be PartFunc of REAL,REAL; ::_thesis: ( f . x0 <> r & f is_differentiable_in x0 implies ex N being Neighbourhood of x0 st
( N c= dom f & ( for g being Real st g in N holds
f . g <> r ) ) )
assume that
A1: f . x0 <> r and
A2: f is_differentiable_in x0 ; ::_thesis: ex N being Neighbourhood of x0 st
( N c= dom f & ( for g being Real st g in N holds
f . g <> r ) )
ex N being Neighbourhood of x0 st
( N c= dom f & ex L being LinearFunc ex R being RestFunc st
for r being Real st r in N holds
(f . r) - (f . x0) = (L . (r - x0)) + (R . (r - x0)) ) by A2, FDIFF_1:def_4;
hence ex N being Neighbourhood of x0 st
( N c= dom f & ( for g being Real st g in N holds
f . g <> r ) ) by A1, A2, FCONT_3:7, FDIFF_1:24; ::_thesis: verum
end;
Lm1: for x0 being Real
for f being PartFunc of REAL,REAL st ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ) holds
( f is_differentiable_in x0 & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
diff (f,x0) = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) ) )
proof
let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ) holds
( f is_differentiable_in x0 & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
diff (f,x0) = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) ) )
let f be PartFunc of REAL,REAL; ::_thesis: ( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ) implies ( f is_differentiable_in x0 & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
diff (f,x0) = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) ) ) )
deffunc H1( Real) -> Element of NAT = 0 ;
defpred S1[ set ] means $1 in REAL ;
given N being Neighbourhood of x0 such that A1: N c= dom f ; ::_thesis: ( ex h being non-zero 0 -convergent Real_Sequence ex c being V8() Real_Sequence st
( rng c = {x0} & rng (h + c) c= dom f & not (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ) or ( f is_differentiable_in x0 & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
diff (f,x0) = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) ) ) )
assume A2: for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ; ::_thesis: ( f is_differentiable_in x0 & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
diff (f,x0) = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) ) )
then A3: for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f & {x0} c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ;
consider r being real number such that
A4: 0 < r and
A5: N = ].(x0 - r),(x0 + r).[ by RCOMP_1:def_6;
consider h being non-zero 0 -convergent Real_Sequence, c being V8() Real_Sequence such that
A6: rng c = {x0} and
A7: rng (h + c) c= dom f and
A8: {x0} c= dom f by A1, Th8;
set l = lim ((h ") (#) ((f /* (h + c)) - (f /* c)));
deffunc H2( Real) -> Element of REAL = (lim ((h ") (#) ((f /* (h + c)) - (f /* c)))) * $1;
consider L being PartFunc of REAL,REAL such that
A9: for g being Real holds
( g in dom L iff S1[g] ) and
A10: for g being Real st g in dom L holds
L . g = H2(g) from SEQ_1:sch_3();
A11: dom L = REAL by A9, FDIFF_1:1;
then A12: for g being Real holds L . g = (lim ((h ") (#) ((f /* (h + c)) - (f /* c)))) * g by A10;
A13: L is total by A11, PARTFUN1:def_2;
deffunc H3( Real) -> Element of REAL = $1 + x0;
consider T being PartFunc of REAL,REAL such that
A14: for g being Real holds
( g in dom T iff S1[g] ) and
A15: for g being Real st g in dom T holds
T . g = H3(g) from SEQ_1:sch_3();
A16: dom T = REAL by A14, FDIFF_1:1;
deffunc H4( real number ) -> Element of REAL = ((f * T) . $1) - (f . x0);
consider T1 being PartFunc of REAL,REAL such that
A17: for g being Real holds
( g in dom T1 iff S1[g] ) and
A18: for g being Real st g in dom T1 holds
T1 . g = H4(g) from SEQ_1:sch_3();
deffunc H5( Real) -> Element of REAL = (T1 - L) . $1;
A19: dom T1 = REAL by A17, FDIFF_1:1;
then A20: T1 is total by PARTFUN1:def_2;
reconsider L = L as LinearFunc by A13, A12, FDIFF_1:def_3;
defpred S2[ set ] means $1 in ].(- r),r.[;
consider R being PartFunc of REAL,REAL such that
A21: R is total and
A22: for g being Real st g in dom R holds
( ( S2[g] implies R . g = H5(g) ) & ( not S2[g] implies R . g = H1(g) ) ) from SCHEME1:sch_8();
A23: dom R = REAL by A21, PARTFUN1:def_2;
A24: now__::_thesis:_for_n_being_Element_of_NAT_holds_c_._n_=_x0
let n be Element of NAT ; ::_thesis: c . n = x0
c . n in {x0} by A6, VALUED_0:28;
hence c . n = x0 by TARSKI:def_1; ::_thesis: verum
end;
now__::_thesis:_for_h1_being_non-zero_0_-convergent_Real_Sequence_holds_
(_(h1_")_(#)_(R_/*_h1)_is_convergent_&_lim_((h1_")_(#)_(R_/*_h1))_=_0_)
let h1 be non-zero 0 -convergent Real_Sequence; ::_thesis: ( (h1 ") (#) (R /* h1) is convergent & lim ((h1 ") (#) (R /* h1)) = 0 )
A25: lim h1 = 0 ;
consider k being Element of NAT such that
A26: for n being Element of NAT st k <= n holds
abs ((h1 . n) - 0) < r by A4, A25, SEQ_2:def_7;
set h2 = h1 ^\ k;
A27: now__::_thesis:_for_n_being_Element_of_NAT_holds_(h1_^\_k)_._n_in_].(-_r),r.[
let n be Element of NAT ; ::_thesis: (h1 ^\ k) . n in ].(- r),r.[
abs ((h1 . (n + k)) - 0) < r by A26, NAT_1:12;
then h1 . (n + k) in ].(0 - r),(0 + r).[ by RCOMP_1:1;
hence (h1 ^\ k) . n in ].(- r),r.[ by NAT_1:def_3; ::_thesis: verum
end;
A28: rng ((h1 ^\ k) + c) c= dom f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng ((h1 ^\ k) + c) or x in dom f )
assume x in rng ((h1 ^\ k) + c) ; ::_thesis: x in dom f
then consider n being Element of NAT such that
A29: x = ((h1 ^\ k) + c) . n by FUNCT_2:113;
(h1 ^\ k) . n in ].(- r),r.[ by A27;
then (h1 ^\ k) . n in { g where g is Real : ( - r < g & g < r ) } by RCOMP_1:def_2;
then A30: ex g being Real st
( g = (h1 ^\ k) . n & - r < g & g < r ) ;
then A31: ((h1 ^\ k) . n) + x0 < x0 + r by XREAL_1:6;
x0 + (- r) < ((h1 ^\ k) . n) + x0 by A30, XREAL_1:6;
then A32: ((h1 ^\ k) . n) + x0 in { g where g is Real : ( x0 - r < g & g < x0 + r ) } by A31;
x = ((h1 ^\ k) . n) + (c . n) by A29, SEQ_1:7
.= ((h1 ^\ k) . n) + x0 by A24 ;
then x in ].(x0 - r),(x0 + r).[ by A32, RCOMP_1:def_2;
hence x in dom f by A1, A5; ::_thesis: verum
end;
set b = ((h1 ^\ k) ") (#) (L /* (h1 ^\ k));
set a = ((h1 ^\ k) ") (#) (T1 /* (h1 ^\ k));
A33: ((h1 ") (#) (R /* h1)) ^\ k = ((h1 ") ^\ k) (#) ((R /* h1) ^\ k) by SEQM_3:19
.= ((h1 ^\ k) ") (#) ((R /* h1) ^\ k) by SEQM_3:18
.= ((h1 ^\ k) ") (#) (R /* (h1 ^\ k)) by A21, VALUED_0:29 ;
A34: (((h1 ^\ k) ") (#) (T1 /* (h1 ^\ k))) - (((h1 ^\ k) ") (#) (L /* (h1 ^\ k))) = ((h1 ^\ k) ") (#) ((T1 /* (h1 ^\ k)) - (L /* (h1 ^\ k))) by SEQ_1:21
.= ((h1 ^\ k) ") (#) ((T1 - L) /* (h1 ^\ k)) by A13, A20, RFUNCT_2:13 ;
A35: now__::_thesis:_for_n_being_Element_of_NAT_holds_((((h1_^\_k)_")_(#)_(T1_/*_(h1_^\_k)))_-_(((h1_^\_k)_")_(#)_(L_/*_(h1_^\_k))))_._n_=_(((h1_^\_k)_")_(#)_(R_/*_(h1_^\_k)))_._n
let n be Element of NAT ; ::_thesis: ((((h1 ^\ k) ") (#) (T1 /* (h1 ^\ k))) - (((h1 ^\ k) ") (#) (L /* (h1 ^\ k)))) . n = (((h1 ^\ k) ") (#) (R /* (h1 ^\ k))) . n
A36: (h1 ^\ k) . n in ].(- r),r.[ by A27;
thus ((((h1 ^\ k) ") (#) (T1 /* (h1 ^\ k))) - (((h1 ^\ k) ") (#) (L /* (h1 ^\ k)))) . n = (((h1 ^\ k) ") . n) * (((T1 - L) /* (h1 ^\ k)) . n) by A34, SEQ_1:8
.= (((h1 ^\ k) ") . n) * ((T1 - L) . ((h1 ^\ k) . n)) by A13, A20, FUNCT_2:115
.= (((h1 ^\ k) ") . n) * (R . ((h1 ^\ k) . n)) by A22, A23, A36
.= (((h1 ^\ k) ") . n) * ((R /* (h1 ^\ k)) . n) by A21, FUNCT_2:115
.= (((h1 ^\ k) ") (#) (R /* (h1 ^\ k))) . n by SEQ_1:8 ; ::_thesis: verum
end;
A38: now__::_thesis:_for_n_being_Nat_holds_(((h1_^\_k)_")_(#)_(L_/*_(h1_^\_k)))_._n_=_lim_((h_")_(#)_((f_/*_(h_+_c))_-_(f_/*_c)))
let n be Nat; ::_thesis: (((h1 ^\ k) ") (#) (L /* (h1 ^\ k))) . n = lim ((h ") (#) ((f /* (h + c)) - (f /* c)))
A39: n in NAT by ORDINAL1:def_12;
then A40: (h1 ^\ k) . n <> 0 by SEQ_1:5;
thus (((h1 ^\ k) ") (#) (L /* (h1 ^\ k))) . n = (((h1 ^\ k) ") . n) * ((L /* (h1 ^\ k)) . n) by A39, SEQ_1:8
.= (((h1 ^\ k) ") . n) * (L . ((h1 ^\ k) . n)) by A13, A39, FUNCT_2:115
.= (((h1 ^\ k) ") . n) * (((h1 ^\ k) . n) * (lim ((h ") (#) ((f /* (h + c)) - (f /* c))))) by A10, A11
.= ((((h1 ^\ k) ") . n) * ((h1 ^\ k) . n)) * (lim ((h ") (#) ((f /* (h + c)) - (f /* c))))
.= ((((h1 ^\ k) . n) ") * ((h1 ^\ k) . n)) * (lim ((h ") (#) ((f /* (h + c)) - (f /* c)))) by VALUED_1:10
.= 1 * (lim ((h ") (#) ((f /* (h + c)) - (f /* c)))) by A40, XCMPLX_0:def_7
.= lim ((h ") (#) ((f /* (h + c)) - (f /* c))) ; ::_thesis: verum
end;
then A41: ((h1 ^\ k) ") (#) (L /* (h1 ^\ k)) is V8() by VALUED_0:def_18;
now__::_thesis:_for_n_being_Element_of_NAT_holds_(((h1_^\_k)_")_(#)_(T1_/*_(h1_^\_k)))_._n_=_(((h1_^\_k)_")_(#)_((f_/*_((h1_^\_k)_+_c))_-_(f_/*_c)))_._n
let n be Element of NAT ; ::_thesis: (((h1 ^\ k) ") (#) (T1 /* (h1 ^\ k))) . n = (((h1 ^\ k) ") (#) ((f /* ((h1 ^\ k) + c)) - (f /* c))) . n
A42: c . n = x0 by A24;
thus (((h1 ^\ k) ") (#) (T1 /* (h1 ^\ k))) . n = (((h1 ^\ k) ") . n) * ((T1 /* (h1 ^\ k)) . n) by SEQ_1:8
.= (((h1 ^\ k) ") . n) * (T1 . ((h1 ^\ k) . n)) by A20, FUNCT_2:115
.= (((h1 ^\ k) ") . n) * (((f * T) . ((h1 ^\ k) . n)) - (f . x0)) by A18, A19
.= (((h1 ^\ k) ") . n) * ((f . (T . ((h1 ^\ k) . n))) - (f . x0)) by A16, FUNCT_1:13
.= (((h1 ^\ k) ") . n) * ((f . (((h1 ^\ k) . n) + x0)) - (f . x0)) by A15, A16
.= (((h1 ^\ k) ") . n) * ((f . (((h1 ^\ k) + c) . n)) - (f . (c . n))) by A42, SEQ_1:7
.= (((h1 ^\ k) ") . n) * ((f . (((h1 ^\ k) + c) . n)) - ((f /* c) . n)) by A6, A8, FUNCT_2:108
.= (((h1 ^\ k) ") . n) * (((f /* ((h1 ^\ k) + c)) . n) - ((f /* c) . n)) by A28, FUNCT_2:108
.= (((h1 ^\ k) ") . n) * (((f /* ((h1 ^\ k) + c)) - (f /* c)) . n) by RFUNCT_2:1
.= (((h1 ^\ k) ") (#) ((f /* ((h1 ^\ k) + c)) - (f /* c))) . n by SEQ_1:8 ; ::_thesis: verum
end;
then A43: ((h1 ^\ k) ") (#) (T1 /* (h1 ^\ k)) = ((h1 ^\ k) ") (#) ((f /* ((h1 ^\ k) + c)) - (f /* c)) by FUNCT_2:63;
then A44: ((h1 ^\ k) ") (#) (T1 /* (h1 ^\ k)) is convergent by A2, A6, A28;
then (((h1 ^\ k) ") (#) (T1 /* (h1 ^\ k))) - (((h1 ^\ k) ") (#) (L /* (h1 ^\ k))) is convergent by A41, SEQ_2:11;
then A45: ((h1 ^\ k) ") (#) (R /* (h1 ^\ k)) is convergent by A35, FUNCT_2:63;
hence (h1 ") (#) (R /* h1) is convergent by A33, SEQ_4:21; ::_thesis: lim ((h1 ") (#) (R /* h1)) = 0
A46: lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = lim (((h1 ^\ k) ") (#) (T1 /* (h1 ^\ k))) by A3, A6, A7, A8, A28, A43, Th7;
lim (((h1 ^\ k) ") (#) (L /* (h1 ^\ k))) = (((h1 ^\ k) ") (#) (L /* (h1 ^\ k))) . 0 by A41, SEQ_4:25
.= lim ((h ") (#) ((f /* (h + c)) - (f /* c))) by A38 ;
then lim ((((h1 ^\ k) ") (#) (T1 /* (h1 ^\ k))) - (((h1 ^\ k) ") (#) (L /* (h1 ^\ k)))) = (lim ((h ") (#) ((f /* (h + c)) - (f /* c)))) - (lim ((h ") (#) ((f /* (h + c)) - (f /* c)))) by A44, A46, A41, SEQ_2:12
.= 0 ;
then lim (((h1 ^\ k) ") (#) (R /* (h1 ^\ k))) = 0 by A35, FUNCT_2:63;
hence lim ((h1 ") (#) (R /* h1)) = 0 by A45, A33, SEQ_4:22; ::_thesis: verum
end;
then reconsider R = R as RestFunc by A21, FDIFF_1:def_2;
A47: now__::_thesis:_ex_N_being_Neighbourhood_of_x0_st_
(_N_c=_dom_f_&_ex_L_being_LinearFunc_ex_R_being_RestFunc_st_
(_L_._1_=_lim_((h_")_(#)_((f_/*_(h_+_c))_-_(f_/*_c)))_&_(_for_g1_being_Real_st_g1_in_N_holds_
(L_._(g1_-_x0))_+_(R_._(g1_-_x0))_=_(f_._g1)_-_(f_._x0)_)_)_)
take N = N; ::_thesis: ( N c= dom f & ex L being LinearFunc ex R being RestFunc st
( L . 1 = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) & ( for g1 being Real st g1 in N holds
(L . (g1 - x0)) + (R . (g1 - x0)) = (f . g1) - (f . x0) ) ) )
thus N c= dom f by A1; ::_thesis: ex L being LinearFunc ex R being RestFunc st
( L . 1 = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) & ( for g1 being Real st g1 in N holds
(L . (g1 - x0)) + (R . (g1 - x0)) = (f . g1) - (f . x0) ) )
take L = L; ::_thesis: ex R being RestFunc st
( L . 1 = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) & ( for g1 being Real st g1 in N holds
(L . (g1 - x0)) + (R . (g1 - x0)) = (f . g1) - (f . x0) ) )
take R = R; ::_thesis: ( L . 1 = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) & ( for g1 being Real st g1 in N holds
(L . (g1 - x0)) + (R . (g1 - x0)) = (f . g1) - (f . x0) ) )
thus L . 1 = (lim ((h ") (#) ((f /* (h + c)) - (f /* c)))) * 1 by A10, A11
.= lim ((h ") (#) ((f /* (h + c)) - (f /* c))) ; ::_thesis: for g1 being Real st g1 in N holds
(L . (g1 - x0)) + (R . (g1 - x0)) = (f . g1) - (f . x0)
let g1 be Real; ::_thesis: ( g1 in N implies (L . (g1 - x0)) + (R . (g1 - x0)) = (f . g1) - (f . x0) )
assume g1 in N ; ::_thesis: (L . (g1 - x0)) + (R . (g1 - x0)) = (f . g1) - (f . x0)
then g1 - x0 in ].(- r),r.[ by A5, FCONT_3:2;
hence (L . (g1 - x0)) + (R . (g1 - x0)) = (L . (g1 - x0)) + ((T1 - L) . (g1 - x0)) by A22, A23
.= (L . (g1 - x0)) + ((T1 . (g1 - x0)) - (L . (g1 - x0))) by A13, A20, RFUNCT_1:56
.= ((f * T) . (g1 - x0)) - (f . x0) by A18, A19
.= (f . (T . (g1 - x0))) - (f . x0) by A16, FUNCT_1:13
.= (f . ((g1 - x0) + x0)) - (f . x0) by A15, A16
.= (f . g1) - (f . x0) ;
::_thesis: verum
end;
thus f is_differentiable_in x0 ::_thesis: for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
diff (f,x0) = lim ((h ") (#) ((f /* (h + c)) - (f /* c)))
proof
consider N1 being Neighbourhood of x0 such that
A48: N1 c= dom f and
A49: ex L being LinearFunc ex R being RestFunc st
( L . 1 = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) & ( for g being Real st g in N1 holds
(f . g) - (f . x0) = (L . (g - x0)) + (R . (g - x0)) ) ) by A47;
take N1 ; :: according to FDIFF_1:def_4 ::_thesis: ( N1 c= dom f & ex b1 being Element of K19(K20(REAL,REAL)) ex b2 being Element of K19(K20(REAL,REAL)) st
for b3 being Element of REAL holds
( not b3 in N1 or (f . b3) - (f . x0) = (b1 . (b3 - x0)) + (b2 . (b3 - x0)) ) )
thus N1 c= dom f by A48; ::_thesis: ex b1 being Element of K19(K20(REAL,REAL)) ex b2 being Element of K19(K20(REAL,REAL)) st
for b3 being Element of REAL holds
( not b3 in N1 or (f . b3) - (f . x0) = (b1 . (b3 - x0)) + (b2 . (b3 - x0)) )
consider L1 being LinearFunc, R1 being RestFunc such that
L1 . 1 = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) and
A50: for g being Real st g in N1 holds
(f . g) - (f . x0) = (L1 . (g - x0)) + (R1 . (g - x0)) by A49;
take L1 ; ::_thesis: ex b1 being Element of K19(K20(REAL,REAL)) st
for b2 being Element of REAL holds
( not b2 in N1 or (f . b2) - (f . x0) = (L1 . (b2 - x0)) + (b1 . (b2 - x0)) )
take R1 ; ::_thesis: for b1 being Element of REAL holds
( not b1 in N1 or (f . b1) - (f . x0) = (L1 . (b1 - x0)) + (R1 . (b1 - x0)) )
thus for b1 being Element of REAL holds
( not b1 in N1 or (f . b1) - (f . x0) = (L1 . (b1 - x0)) + (R1 . (b1 - x0)) ) by A50; ::_thesis: verum
end;
then A51: diff (f,x0) = lim ((h ") (#) ((f /* (h + c)) - (f /* c))) by A47, FDIFF_1:def_5;
let h1 be non-zero 0 -convergent Real_Sequence; ::_thesis: for c being V8() Real_Sequence st rng c = {x0} & rng (h1 + c) c= dom f holds
diff (f,x0) = lim ((h1 ") (#) ((f /* (h1 + c)) - (f /* c)))
let c1 be V8() Real_Sequence; ::_thesis: ( rng c1 = {x0} & rng (h1 + c1) c= dom f implies diff (f,x0) = lim ((h1 ") (#) ((f /* (h1 + c1)) - (f /* c1))) )
assume that
A52: rng c1 = {x0} and
A53: rng (h1 + c1) c= dom f ; ::_thesis: diff (f,x0) = lim ((h1 ") (#) ((f /* (h1 + c1)) - (f /* c1)))
c1 = c by A6, A52, Th5;
hence diff (f,x0) = lim ((h1 ") (#) ((f /* (h1 + c1)) - (f /* c1))) by A3, A6, A7, A8, A51, A53, Th7; ::_thesis: verum
end;
theorem Th11: :: FDIFF_2:11
for x0 being Real
for f being PartFunc of REAL,REAL holds
( f is_differentiable_in x0 iff ( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ) ) )
proof
let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL holds
( f is_differentiable_in x0 iff ( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ) ) )
let f be PartFunc of REAL,REAL; ::_thesis: ( f is_differentiable_in x0 iff ( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ) ) )
thus ( f is_differentiable_in x0 implies ( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ) ) ) ::_thesis: ( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ) implies f is_differentiable_in x0 )
proof
assume A1: f is_differentiable_in x0 ; ::_thesis: ( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ) )
then consider N being Neighbourhood of x0 such that
A2: N c= dom f and
ex L being LinearFunc ex R being RestFunc st
for g being Real st g in N holds
(f . g) - (f . x0) = (L . (g - x0)) + (R . (g - x0)) by FDIFF_1:def_4;
thus ex N being Neighbourhood of x0 st N c= dom f by A2; ::_thesis: for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent
let c be V8() Real_Sequence; ::_thesis: ( rng c = {x0} & rng (h + c) c= dom f implies (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent )
assume that
A3: rng c = {x0} and
A4: rng (h + c) c= dom f ; ::_thesis: (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent
A5: lim h = 0 ;
consider r being real number such that
A6: 0 < r and
A7: N = ].(x0 - r),(x0 + r).[ by RCOMP_1:def_6;
consider k being Element of NAT such that
A8: for n being Element of NAT st k <= n holds
abs ((h . n) - 0) < r by A5, A6, SEQ_2:def_7;
set h1 = h ^\ k;
rng ((h ^\ k) + c) c= N
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng ((h ^\ k) + c) or x in N )
assume x in rng ((h ^\ k) + c) ; ::_thesis: x in N
then consider n being Element of NAT such that
A9: x = ((h ^\ k) + c) . n by FUNCT_2:113;
c . n in rng c by VALUED_0:28;
then c . n = x0 by A3, TARSKI:def_1;
then A10: x = ((h ^\ k) . n) + x0 by A9, SEQ_1:7
.= (h . (n + k)) + x0 by NAT_1:def_3 ;
abs ((h . (n + k)) - 0) < r by A8, NAT_1:12;
then h . (n + k) in ].(0 - r),(0 + r).[ by RCOMP_1:1;
then h . (n + k) in { g where g is Real : ( - r < g & g < r ) } by RCOMP_1:def_2;
then A11: ex g being Real st
( g = h . (n + k) & - r < g & g < r ) ;
then A12: (h . (n + k)) + x0 < x0 + r by XREAL_1:6;
x0 + (- r) < (h . (n + k)) + x0 by A11, XREAL_1:6;
then (h . (n + k)) + x0 in { g where g is Real : ( x0 - r < g & g < x0 + r ) } by A12;
hence x in N by A7, A10, RCOMP_1:def_2; ::_thesis: verum
end;
then A13: ((h ^\ k) ") (#) ((f /* ((h ^\ k) + c)) - (f /* c)) is convergent by A1, A2, A3, FDIFF_1:12;
A14: {x0} c= dom f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {x0} or x in dom f )
assume x in {x0} ; ::_thesis: x in dom f
then A15: x = x0 by TARSKI:def_1;
x0 in N by RCOMP_1:16;
hence x in dom f by A2, A15; ::_thesis: verum
end;
c ^\ k = c by VALUED_0:26;
then ((h ^\ k) ") (#) ((f /* ((h ^\ k) + c)) - (f /* c)) = ((h ^\ k) ") (#) ((f /* ((h + c) ^\ k)) - (f /* (c ^\ k))) by SEQM_3:15
.= ((h ^\ k) ") (#) (((f /* (h + c)) ^\ k) - (f /* (c ^\ k))) by A4, VALUED_0:27
.= ((h ^\ k) ") (#) (((f /* (h + c)) ^\ k) - ((f /* c) ^\ k)) by A3, A14, VALUED_0:27
.= ((h ^\ k) ") (#) (((f /* (h + c)) - (f /* c)) ^\ k) by SEQM_3:17
.= ((h ") ^\ k) (#) (((f /* (h + c)) - (f /* c)) ^\ k) by SEQM_3:18
.= ((h ") (#) ((f /* (h + c)) - (f /* c))) ^\ k by SEQM_3:19 ;
hence (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent by A13, SEQ_4:21; ::_thesis: verum
end;
assume that
A16: ex N being Neighbourhood of x0 st N c= dom f and
A17: for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ; ::_thesis: f is_differentiable_in x0
thus f is_differentiable_in x0 by A16, A17, Lm1; ::_thesis: verum
end;
theorem Th12: :: FDIFF_2:12
for x0, g being Real
for f being PartFunc of REAL,REAL holds
( f is_differentiable_in x0 & diff (f,x0) = g iff ( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g ) ) ) )
proof
let x0, g be Real; ::_thesis: for f being PartFunc of REAL,REAL holds
( f is_differentiable_in x0 & diff (f,x0) = g iff ( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g ) ) ) )
let f be PartFunc of REAL,REAL; ::_thesis: ( f is_differentiable_in x0 & diff (f,x0) = g iff ( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g ) ) ) )
thus ( f is_differentiable_in x0 & diff (f,x0) = g implies ( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g ) ) ) ) ::_thesis: ( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g ) ) implies ( f is_differentiable_in x0 & diff (f,x0) = g ) )
proof
assume that
A1: f is_differentiable_in x0 and
A2: diff (f,x0) = g ; ::_thesis: ( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g ) ) )
thus ex N being Neighbourhood of x0 st N c= dom f by A1, Th11; ::_thesis: for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g )
A3: for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent by A1, Th11;
ex N being Neighbourhood of x0 st N c= dom f by A1, Th11;
hence for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g ) by A2, A3, Lm1; ::_thesis: verum
end;
assume that
A4: ex N being Neighbourhood of x0 st N c= dom f and
A5: for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g ) ; ::_thesis: ( f is_differentiable_in x0 & diff (f,x0) = g )
A6: for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent by A5;
hence f is_differentiable_in x0 by A4, Lm1; ::_thesis: diff (f,x0) = g
consider h being non-zero 0 -convergent Real_Sequence, c being V8() Real_Sequence such that
A7: rng c = {x0} and
A8: rng (h + c) c= dom f and
{x0} c= dom f by A4, Th8;
lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g by A5, A7, A8;
hence diff (f,x0) = g by A4, A6, A7, A8, Lm1; ::_thesis: verum
end;
Lm2: for x0 being Real
for f2, f1 being PartFunc of REAL,REAL st ex N being Neighbourhood of x0 st N c= dom (f2 * f1) & f1 is_differentiable_in x0 & f2 is_differentiable_in f1 . x0 holds
( f2 * f1 is_differentiable_in x0 & diff ((f2 * f1),x0) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) )
proof
let x0 be Real; ::_thesis: for f2, f1 being PartFunc of REAL,REAL st ex N being Neighbourhood of x0 st N c= dom (f2 * f1) & f1 is_differentiable_in x0 & f2 is_differentiable_in f1 . x0 holds
( f2 * f1 is_differentiable_in x0 & diff ((f2 * f1),x0) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) )
let f2, f1 be PartFunc of REAL,REAL; ::_thesis: ( ex N being Neighbourhood of x0 st N c= dom (f2 * f1) & f1 is_differentiable_in x0 & f2 is_differentiable_in f1 . x0 implies ( f2 * f1 is_differentiable_in x0 & diff ((f2 * f1),x0) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) ) )
given N being Neighbourhood of x0 such that A1: N c= dom (f2 * f1) ; ::_thesis: ( not f1 is_differentiable_in x0 or not f2 is_differentiable_in f1 . x0 or ( f2 * f1 is_differentiable_in x0 & diff ((f2 * f1),x0) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) ) )
assume that
A2: f1 is_differentiable_in x0 and
A3: f2 is_differentiable_in f1 . x0 ; ::_thesis: ( f2 * f1 is_differentiable_in x0 & diff ((f2 * f1),x0) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) )
for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom (f2 * f1) holds
( (h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) is convergent & lim ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) )
proof
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom (f2 * f1) holds
( (h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) is convergent & lim ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) )
let c be V8() Real_Sequence; ::_thesis: ( rng c = {x0} & rng (h + c) c= dom (f2 * f1) implies ( (h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) is convergent & lim ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) ) )
assume that
A4: rng c = {x0} and
A5: rng (h + c) c= dom (f2 * f1) ; ::_thesis: ( (h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) is convergent & lim ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) )
A6: rng (h + c) c= dom f1 by A5, Th9;
set a = f1 /* c;
A7: now__::_thesis:_for_n_being_Element_of_NAT_holds_c_._n_=_x0
let n be Element of NAT ; ::_thesis: c . n = x0
c . n in rng c by VALUED_0:28;
hence c . n = x0 by A4, TARSKI:def_1; ::_thesis: verum
end;
A8: rng c c= dom (f2 * f1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng c or x in dom (f2 * f1) )
assume x in rng c ; ::_thesis: x in dom (f2 * f1)
then A9: ex n being Element of NAT st c . n = x by FUNCT_2:113;
x0 in N by RCOMP_1:16;
then x0 in dom (f2 * f1) by A1;
hence x in dom (f2 * f1) by A7, A9; ::_thesis: verum
end;
set d = (f1 /* (h + c)) - (f1 /* c);
A10: lim h = 0 ;
A12: h + c is convergent by SEQ_2:5;
lim c = c . 0 by SEQ_4:25;
then lim c = x0 by A7;
then A13: lim (h + c) = 0 + x0 by A10, SEQ_2:6
.= x0 ;
A14: f1 is_continuous_in x0 by A2, FDIFF_1:24;
then A15: f1 /* (h + c) is convergent by A6, A12, A13, FCONT_1:def_1;
A16: f1 . x0 = lim (f1 /* (h + c)) by A6, A12, A13, A14, FCONT_1:def_1;
A17: rng (f1 /* (h + c)) c= dom f2 by A5, Th9;
A18: rng c c= dom f1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng c or x in dom f1 )
assume x in rng c ; ::_thesis: x in dom f1
then A19: x = x0 by A4, TARSKI:def_1;
x0 in N by RCOMP_1:16;
hence x in dom f1 by A1, A19, FUNCT_1:11; ::_thesis: verum
end;
A20: rng (f1 /* c) = {(f1 . x0)}
proof
thus rng (f1 /* c) c= {(f1 . x0)} :: according to XBOOLE_0:def_10 ::_thesis: {(f1 . x0)} c= rng (f1 /* c)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* c) or x in {(f1 . x0)} )
assume x in rng (f1 /* c) ; ::_thesis: x in {(f1 . x0)}
then consider n being Element of NAT such that
A21: (f1 /* c) . n = x by FUNCT_2:113;
c . n = x0 by A7;
then x = f1 . x0 by A18, A21, FUNCT_2:108;
hence x in {(f1 . x0)} by TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(f1 . x0)} or x in rng (f1 /* c) )
A22: (f1 /* c) . 0 in rng (f1 /* c) by VALUED_0:28;
assume x in {(f1 . x0)} ; ::_thesis: x in rng (f1 /* c)
then A23: x = f1 . x0 by TARSKI:def_1;
c . 0 = x0 by A7;
hence x in rng (f1 /* c) by A18, A23, A22, FUNCT_2:108; ::_thesis: verum
end;
A24: rng (f1 /* c) c= dom f2
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (f1 /* c) or x in dom f2 )
assume x in rng (f1 /* c) ; ::_thesis: x in dom f2
then A25: x = f1 . x0 by A20, TARSKI:def_1;
x0 in N by RCOMP_1:16;
hence x in dom f2 by A1, A25, FUNCT_1:11; ::_thesis: verum
end;
A26: now__::_thesis:_for_n_being_Nat_holds_(f1_/*_c)_._n_=_(f1_/*_c)_._(n_+_1)
let n be Nat; ::_thesis: (f1 /* c) . n = (f1 /* c) . (n + 1)
(f1 /* c) . n in rng (f1 /* c) by VALUED_0:28;
then A27: (f1 /* c) . n = f1 . x0 by A20, TARSKI:def_1;
(f1 /* c) . (n + 1) in rng (f1 /* c) by VALUED_0:28;
hence (f1 /* c) . n = (f1 /* c) . (n + 1) by A20, A27, TARSKI:def_1; ::_thesis: verum
end;
then f1 /* c is V8() by VALUED_0:25;
then A28: (f1 /* (h + c)) - (f1 /* c) is convergent by A15, SEQ_2:11;
reconsider a = f1 /* c as V8() Real_Sequence by A26, VALUED_0:25;
a . 0 in rng a by VALUED_0:28;
then a . 0 = f1 . x0 by A20, TARSKI:def_1;
then lim a = f1 . x0 by SEQ_4:25;
then A29: lim ((f1 /* (h + c)) - (f1 /* c)) = (f1 . x0) - (f1 . x0) by A15, A16, SEQ_2:12
.= 0 ;
( (h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) is convergent & lim ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) )
proof
percases ( ex k being Element of NAT st
for n being Element of NAT st k <= n holds
(f1 /* (h + c)) . n <> f1 . x0 or for k being Element of NAT ex n being Element of NAT st
( k <= n & (f1 /* (h + c)) . n = f1 . x0 ) ) ;
suppose ex k being Element of NAT st
for n being Element of NAT st k <= n holds
(f1 /* (h + c)) . n <> f1 . x0 ; ::_thesis: ( (h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) is convergent & lim ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) )
then consider k being Element of NAT such that
A30: for n being Element of NAT st k <= n holds
(f1 /* (h + c)) . n <> f1 . x0 ;
now__::_thesis:_for_n_being_Element_of_NAT_holds_not_(((f1_/*_(h_+_c))_-_(f1_/*_c))_^\_k)_._n_=_0
given n being Element of NAT such that A31: (((f1 /* (h + c)) - (f1 /* c)) ^\ k) . n = 0 ; ::_thesis: contradiction
0 = ((f1 /* (h + c)) - (f1 /* c)) . (n + k) by A31, NAT_1:def_3
.= ((f1 /* (h + c)) . (n + k)) - ((f1 /* c) . (n + k)) by RFUNCT_2:1
.= ((f1 /* (h + c)) . (n + k)) - (f1 . (c . (n + k))) by A18, FUNCT_2:108
.= ((f1 /* (h + c)) . (n + k)) - (f1 . x0) by A7 ;
hence contradiction by A30, NAT_1:12; ::_thesis: verum
end;
then A32: ((f1 /* (h + c)) - (f1 /* c)) ^\ k is non-zero by SEQ_1:5;
set c1 = c ^\ k;
set h1 = h ^\ k;
set a1 = a ^\ k;
set s = ((h ^\ k) ") (#) ((f1 /* ((h ^\ k) + (c ^\ k))) - (f1 /* (c ^\ k)));
A33: now__::_thesis:_for_n_being_Element_of_NAT_holds_(((f1_/*_(h_+_c))_-_(f1_/*_c))_+_a)_._n_=_(f1_/*_(h_+_c))_._n
let n be Element of NAT ; ::_thesis: (((f1 /* (h + c)) - (f1 /* c)) + a) . n = (f1 /* (h + c)) . n
thus (((f1 /* (h + c)) - (f1 /* c)) + a) . n = (((f1 /* (h + c)) - (f1 /* c)) . n) + (a . n) by SEQ_1:7
.= (((f1 /* (h + c)) . n) - (a . n)) + (a . n) by RFUNCT_2:1
.= (f1 /* (h + c)) . n ; ::_thesis: verum
end;
lim (((f1 /* (h + c)) - (f1 /* c)) ^\ k) = 0 by A28, A29, SEQ_4:20;
then reconsider d1 = ((f1 /* (h + c)) - (f1 /* c)) ^\ k as non-zero 0 -convergent Real_Sequence by A28, A32, FDIFF_1:def_1;
set t = (d1 ") (#) ((f2 /* (d1 + (a ^\ k))) - (f2 /* (a ^\ k)));
d1 + (a ^\ k) = (((f1 /* (h + c)) - (f1 /* c)) + a) ^\ k by SEQM_3:15;
then A34: d1 + (a ^\ k) = (f1 /* (h + c)) ^\ k by A33, FUNCT_2:63;
rng ((f1 /* (h + c)) ^\ k) c= rng (f1 /* (h + c)) by VALUED_0:21;
then A35: rng (d1 + (a ^\ k)) c= dom f2 by A17, A34, XBOOLE_1:1;
A36: rng (a ^\ k) = {(f1 . x0)} by A20, VALUED_0:26;
then A37: lim ((d1 ") (#) ((f2 /* (d1 + (a ^\ k))) - (f2 /* (a ^\ k)))) = diff (f2,(f1 . x0)) by A3, A35, Th12;
diff (f2,(f1 . x0)) = diff (f2,(f1 . x0)) ;
then A38: (d1 ") (#) ((f2 /* (d1 + (a ^\ k))) - (f2 /* (a ^\ k))) is convergent by A3, A36, A35, Th12;
rng ((h + c) ^\ k) c= rng (h + c) by VALUED_0:21;
then rng ((h + c) ^\ k) c= dom f1 by A6, XBOOLE_1:1;
then A39: rng ((h ^\ k) + (c ^\ k)) c= dom f1 by SEQM_3:15;
A40: rng (c ^\ k) = {x0} by A4, VALUED_0:26;
diff (f1,x0) = diff (f1,x0) ;
then A41: ((h ^\ k) ") (#) ((f1 /* ((h ^\ k) + (c ^\ k))) - (f1 /* (c ^\ k))) is convergent by A2, A40, A39, Th12;
A42: ((d1 ") (#) ((f2 /* (d1 + (a ^\ k))) - (f2 /* (a ^\ k)))) (#) (((h ^\ k) ") (#) ((f1 /* ((h ^\ k) + (c ^\ k))) - (f1 /* (c ^\ k)))) = ((d1 ") (#) ((f2 /* (d1 + (a ^\ k))) - (f2 /* (a ^\ k)))) (#) (((f1 /* ((h + c) ^\ k)) - (f1 /* (c ^\ k))) (#) ((h ^\ k) ")) by SEQM_3:15
.= ((d1 ") (#) ((f2 /* (d1 + (a ^\ k))) - (f2 /* (a ^\ k)))) (#) ((((f1 /* (h + c)) ^\ k) - (f1 /* (c ^\ k))) (#) ((h ^\ k) ")) by A6, VALUED_0:27
.= ((d1 ") (#) ((f2 /* (d1 + (a ^\ k))) - (f2 /* (a ^\ k)))) (#) ((((f1 /* (h + c)) ^\ k) - ((f1 /* c) ^\ k)) (#) ((h ^\ k) ")) by A18, VALUED_0:27
.= (((f2 /* (d1 + (a ^\ k))) - (f2 /* (a ^\ k))) /" d1) (#) (d1 (#) ((h ^\ k) ")) by SEQM_3:17
.= ((((f2 /* (d1 + (a ^\ k))) - (f2 /* (a ^\ k))) /" d1) (#) d1) (#) ((h ^\ k) ") by SEQ_1:14
.= ((f2 /* (d1 + (a ^\ k))) - (f2 /* (a ^\ k))) (#) ((h ^\ k) ") by SEQ_1:37
.= (((f2 /* (f1 /* (h + c))) ^\ k) - (f2 /* (a ^\ k))) (#) ((h ^\ k) ") by A17, A34, VALUED_0:27
.= ((((f2 * f1) /* (h + c)) ^\ k) - (f2 /* (a ^\ k))) (#) ((h ^\ k) ") by A5, VALUED_0:31
.= ((((f2 * f1) /* (h + c)) ^\ k) - ((f2 /* a) ^\ k)) (#) ((h ^\ k) ") by A24, VALUED_0:27
.= ((((f2 * f1) /* (h + c)) ^\ k) - (((f2 * f1) /* c) ^\ k)) (#) ((h ^\ k) ") by A8, VALUED_0:31
.= ((((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) ^\ k) (#) ((h ^\ k) ") by SEQM_3:17
.= ((((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) ^\ k) (#) ((h ") ^\ k) by SEQM_3:18
.= ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) ^\ k by SEQM_3:19 ;
then A43: ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) ^\ k is convergent by A41, A38, SEQ_2:14;
hence (h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) is convergent by SEQ_4:21; ::_thesis: lim ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) = (diff (f2,(f1 . x0))) * (diff (f1,x0))
lim (((h ^\ k) ") (#) ((f1 /* ((h ^\ k) + (c ^\ k))) - (f1 /* (c ^\ k)))) = diff (f1,x0) by A2, A40, A39, Th12;
then lim (((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) ^\ k) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) by A42, A41, A38, A37, SEQ_2:15;
hence lim ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) by A43, SEQ_4:22; ::_thesis: verum
end;
supposeA44: for k being Element of NAT ex n being Element of NAT st
( k <= n & (f1 /* (h + c)) . n = f1 . x0 ) ; ::_thesis: ( (h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) is convergent & lim ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) )
defpred S1[ set ] means $1 = f1 . x0;
A45: for k being Element of NAT ex n being Element of NAT st
( k <= n & S1[(f1 /* (h + c)) . n] ) by A44;
consider I1 being V37() sequence of NAT such that
A46: for n being Element of NAT holds S1[((f1 /* (h + c)) * I1) . n] and
for n being Element of NAT st ( for r being Real st r = (f1 /* (h + c)) . n holds
S1[r] ) holds
ex m being Element of NAT st n = I1 . m from FDIFF_2:sch_1(A45);
h * I1 is subsequence of h by VALUED_0:def_17;
then reconsider h1 = h * I1 as non-zero 0 -convergent Real_Sequence by Th6;
set c1 = c * I1;
A47: c * I1 is subsequence of c by VALUED_0:def_17;
then A48: c * I1 = c by VALUED_0:26;
A49: rng (c * I1) = {x0} by A4, A47, VALUED_0:26;
reconsider c1 = c * I1 as V8() Real_Sequence ;
A50: now__::_thesis:_for_g_being_real_number_st_0_<_g_holds_
ex_n_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n_<=_m_holds_
abs_((((h1_")_(#)_((f1_/*_(h1_+_c1))_-_(f1_/*_c1)))_._m)_-_0)_<_g
let g be real number ; ::_thesis: ( 0 < g implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((((h1 ") (#) ((f1 /* (h1 + c1)) - (f1 /* c1))) . m) - 0) < g )
assume A51: 0 < g ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((((h1 ") (#) ((f1 /* (h1 + c1)) - (f1 /* c1))) . m) - 0) < g
take n = 0 ; ::_thesis: for m being Element of NAT st n <= m holds
abs ((((h1 ") (#) ((f1 /* (h1 + c1)) - (f1 /* c1))) . m) - 0) < g
let m be Element of NAT ; ::_thesis: ( n <= m implies abs ((((h1 ") (#) ((f1 /* (h1 + c1)) - (f1 /* c1))) . m) - 0) < g )
assume n <= m ; ::_thesis: abs ((((h1 ") (#) ((f1 /* (h1 + c1)) - (f1 /* c1))) . m) - 0) < g
abs ((((h1 ") (#) ((f1 /* (h1 + c1)) - (f1 /* c1))) . m) - 0) = abs (((h1 ") . m) * (((f1 /* (h1 + c1)) - (f1 /* c1)) . m)) by SEQ_1:8
.= abs (((h1 ") . m) * (((f1 /* (h1 + c1)) . m) - ((f1 /* c1) . m))) by RFUNCT_2:1
.= abs (((h1 ") . m) * (((f1 /* (h1 + c1)) . m) - (f1 . (c . m)))) by A18, A48, FUNCT_2:108
.= abs (((h1 ") . m) * (((f1 /* (h1 + c1)) . m) - (f1 . x0))) by A7
.= abs (((h1 ") . m) * (((f1 /* ((h + c) * I1)) . m) - (f1 . x0))) by RFUNCT_2:2
.= abs (((h1 ") . m) * ((((f1 /* (h + c)) * I1) . m) - (f1 . x0))) by A6, FUNCT_2:110
.= abs (((h1 ") . m) * ((f1 . x0) - (f1 . x0))) by A46
.= 0 by ABSVALUE:2 ;
hence abs ((((h1 ") (#) ((f1 /* (h1 + c1)) - (f1 /* c1))) . m) - 0) < g by A51; ::_thesis: verum
end;
(h + c) * I1 is subsequence of h + c by VALUED_0:def_17;
then rng ((h + c) * I1) c= rng (h + c) by VALUED_0:21;
then rng ((h + c) * I1) c= dom f1 by A6, XBOOLE_1:1;
then A52: rng (h1 + c1) c= dom f1 by RFUNCT_2:2;
then A53: lim ((h1 ") (#) ((f1 /* (h1 + c1)) - (f1 /* c1))) = diff (f1,x0) by A2, A49, Th12;
diff (f1,x0) = diff (f1,x0) ;
then (h1 ") (#) ((f1 /* (h1 + c1)) - (f1 /* c1)) is convergent by A2, A49, A52, Th12;
then A54: diff (f1,x0) = 0 by A53, A50, SEQ_2:def_7;
now__::_thesis:_(_(h_")_(#)_(((f2_*_f1)_/*_(h_+_c))_-_((f2_*_f1)_/*_c))_is_convergent_&_lim_((h_")_(#)_(((f2_*_f1)_/*_(h_+_c))_-_((f2_*_f1)_/*_c)))_=_(diff_(f2,(f1_._x0)))_*_(diff_(f1,x0))_)
percases ( ex k being Element of NAT st
for n being Element of NAT st k <= n holds
(f1 /* (h + c)) . n = f1 . x0 or for k being Element of NAT ex n being Element of NAT st
( k <= n & (f1 /* (h + c)) . n <> f1 . x0 ) ) ;
suppose ex k being Element of NAT st
for n being Element of NAT st k <= n holds
(f1 /* (h + c)) . n = f1 . x0 ; ::_thesis: ( (h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) is convergent & lim ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) )
then consider k being Element of NAT such that
A55: for n being Element of NAT st k <= n holds
(f1 /* (h + c)) . n = f1 . x0 ;
A56: now__::_thesis:_for_g_being_real_number_st_0_<_g_holds_
ex_n_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n_<=_m_holds_
abs_((((h_")_(#)_(((f2_*_f1)_/*_(h_+_c))_-_((f2_*_f1)_/*_c)))_._m)_-_((diff_(f2,(f1_._x0)))_*_(diff_(f1,x0))))_<_g
let g be real number ; ::_thesis: ( 0 < g implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g )
assume A57: 0 < g ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g
take n = k; ::_thesis: for m being Element of NAT st n <= m holds
abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g
let m be Element of NAT ; ::_thesis: ( n <= m implies abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g )
assume n <= m ; ::_thesis: abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g
then A58: (f1 /* (h + c)) . m = f1 . x0 by A55;
abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) = abs (((h ") . m) * ((((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) . m)) by A54, SEQ_1:8
.= abs (((h ") . m) * ((((f2 * f1) /* (h + c)) . m) - (((f2 * f1) /* c) . m))) by RFUNCT_2:1
.= abs (((h ") . m) * (((f2 /* (f1 /* (h + c))) . m) - (((f2 * f1) /* c) . m))) by A5, VALUED_0:31
.= abs (((h ") . m) * ((f2 . ((f1 /* (h + c)) . m)) - (((f2 * f1) /* c) . m))) by A17, FUNCT_2:108
.= abs (((h ") . m) * ((f2 . (f1 . x0)) - ((f2 /* (f1 /* c)) . m))) by A8, A58, VALUED_0:31
.= abs (((h ") . m) * ((f2 . (f1 . x0)) - (f2 . ((f1 /* c) . m)))) by A24, FUNCT_2:108
.= abs (((h ") . m) * ((f2 . (f1 . x0)) - (f2 . (f1 . (c . m))))) by A18, FUNCT_2:108
.= abs (((h ") . m) * ((f2 . (f1 . x0)) - (f2 . (f1 . x0)))) by A7
.= 0 by ABSVALUE:2 ;
hence abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g by A57; ::_thesis: verum
end;
hence (h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) is convergent by SEQ_2:def_6; ::_thesis: lim ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) = (diff (f2,(f1 . x0))) * (diff (f1,x0))
hence lim ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) by A56, SEQ_2:def_7; ::_thesis: verum
end;
supposeA59: for k being Element of NAT ex n being Element of NAT st
( k <= n & (f1 /* (h + c)) . n <> f1 . x0 ) ; ::_thesis: ( (h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) is convergent & lim ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) )
defpred S2[ set ] means $1 <> f1 . x0;
A60: for k being Element of NAT ex n being Element of NAT st
( k <= n & S2[(f1 /* (h + c)) . n] ) by A59;
consider I2 being V37() sequence of NAT such that
A61: for n being Element of NAT holds S2[((f1 /* (h + c)) * I2) . n] and
A62: for n being Element of NAT st ( for r being Real st r = (f1 /* (h + c)) . n holds
S2[r] ) holds
ex m being Element of NAT st n = I2 . m from FDIFF_2:sch_1(A60);
now__::_thesis:_for_n_being_Element_of_NAT_holds_not_(((f1_/*_(h_+_c))_-_(f1_/*_c))_*_I2)_._n_=_0
given n being Element of NAT such that A63: (((f1 /* (h + c)) - (f1 /* c)) * I2) . n = 0 ; ::_thesis: contradiction
0 = ((f1 /* (h + c)) - (f1 /* c)) . (I2 . n) by A63, FUNCT_2:15
.= ((f1 /* (h + c)) . (I2 . n)) - ((f1 /* c) . (I2 . n)) by RFUNCT_2:1
.= ((f1 /* (h + c)) . (I2 . n)) - (f1 . (c . (I2 . n))) by A18, FUNCT_2:108
.= ((f1 /* (h + c)) . (I2 . n)) - (f1 . x0) by A7
.= (((f1 /* (h + c)) * I2) . n) - (f1 . x0) by FUNCT_2:15 ;
hence contradiction by A61; ::_thesis: verum
end;
then A64: ((f1 /* (h + c)) - (f1 /* c)) * I2 is non-zero by SEQ_1:5;
h * I2 is subsequence of h by VALUED_0:def_17;
then reconsider h2 = h * I2 as non-zero 0 -convergent Real_Sequence by Th6;
set a1 = a * I2;
set c2 = c * I2;
reconsider c2 = c * I2 as V8() Real_Sequence ;
set s = (h2 ") (#) ((f1 /* (h2 + c2)) - (f1 /* c2));
A65: ((f1 /* (h + c)) - (f1 /* c)) * I2 is subsequence of (f1 /* (h + c)) - (f1 /* c) by VALUED_0:def_17;
then A66: ((f1 /* (h + c)) - (f1 /* c)) * I2 is convergent by A28, SEQ_4:16;
lim (((f1 /* (h + c)) - (f1 /* c)) * I2) = 0 by A28, A29, A65, SEQ_4:17;
then reconsider d1 = ((f1 /* (h + c)) - (f1 /* c)) * I2 as non-zero 0 -convergent Real_Sequence by A66, A64, FDIFF_1:def_1;
set t = (d1 ") (#) ((f2 /* (d1 + (a * I2))) - (f2 /* (a * I2)));
a * I2 is subsequence of a by VALUED_0:def_17;
then A67: rng (a * I2) = {(f1 . x0)} by A20, VALUED_0:26;
A68: now__::_thesis:_for_n_being_Element_of_NAT_holds_(((f1_/*_(h_+_c))_-_(f1_/*_c))_+_a)_._n_=_(f1_/*_(h_+_c))_._n
let n be Element of NAT ; ::_thesis: (((f1 /* (h + c)) - (f1 /* c)) + a) . n = (f1 /* (h + c)) . n
thus (((f1 /* (h + c)) - (f1 /* c)) + a) . n = (((f1 /* (h + c)) - (f1 /* c)) . n) + (a . n) by SEQ_1:7
.= (((f1 /* (h + c)) . n) - (a . n)) + (a . n) by RFUNCT_2:1
.= (f1 /* (h + c)) . n ; ::_thesis: verum
end;
d1 + (a * I2) = (((f1 /* (h + c)) - (f1 /* c)) + a) * I2 by RFUNCT_2:2;
then A69: d1 + (a * I2) = (f1 /* (h + c)) * I2 by A68, FUNCT_2:63;
A70: ((d1 ") (#) ((f2 /* (d1 + (a * I2))) - (f2 /* (a * I2)))) (#) ((h2 ") (#) ((f1 /* (h2 + c2)) - (f1 /* c2))) = ((d1 ") (#) ((f2 /* (d1 + (a * I2))) - (f2 /* (a * I2)))) (#) (((f1 /* ((h + c) * I2)) - (f1 /* c2)) (#) (h2 ")) by RFUNCT_2:2
.= ((d1 ") (#) ((f2 /* (d1 + (a * I2))) - (f2 /* (a * I2)))) (#) ((((f1 /* (h + c)) * I2) - (f1 /* c2)) (#) (h2 ")) by A6, FUNCT_2:110
.= ((d1 ") (#) ((f2 /* (d1 + (a * I2))) - (f2 /* (a * I2)))) (#) ((((f1 /* (h + c)) * I2) - ((f1 /* c) * I2)) (#) (h2 ")) by A18, FUNCT_2:110
.= (((f2 /* (d1 + (a * I2))) - (f2 /* (a * I2))) /" d1) (#) (d1 (#) (h2 ")) by RFUNCT_2:2
.= ((((f2 /* (d1 + (a * I2))) - (f2 /* (a * I2))) /" d1) (#) d1) (#) (h2 ") by SEQ_1:14
.= ((f2 /* (d1 + (a * I2))) - (f2 /* (a * I2))) (#) (h2 ") by SEQ_1:37
.= (((f2 /* (f1 /* (h + c))) * I2) - (f2 /* (a * I2))) (#) (h2 ") by A17, A69, FUNCT_2:110
.= ((((f2 * f1) /* (h + c)) * I2) - (f2 /* (a * I2))) (#) (h2 ") by A5, VALUED_0:31
.= ((((f2 * f1) /* (h + c)) * I2) - ((f2 /* a) * I2)) (#) (h2 ") by A24, FUNCT_2:110
.= ((((f2 * f1) /* (h + c)) * I2) - (((f2 * f1) /* c) * I2)) (#) (h2 ") by A8, VALUED_0:31
.= ((((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) * I2) (#) (h2 ") by RFUNCT_2:2
.= ((((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) * I2) (#) ((h ") * I2) by RFUNCT_2:5
.= ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) * I2 by RFUNCT_2:2 ;
reconsider a1 = a * I2 as V8() Real_Sequence ;
(f1 /* (h + c)) * I2 is subsequence of f1 /* (h + c) by VALUED_0:def_17;
then rng ((f1 /* (h + c)) * I2) c= rng (f1 /* (h + c)) by VALUED_0:21;
then A71: rng (d1 + a1) c= dom f2 by A17, A69, XBOOLE_1:1;
(h + c) * I2 is subsequence of h + c by VALUED_0:def_17;
then rng ((h + c) * I2) c= rng (h + c) by VALUED_0:21;
then rng ((h + c) * I2) c= dom f1 by A6, XBOOLE_1:1;
then A72: rng (h2 + c2) c= dom f1 by RFUNCT_2:2;
c2 is subsequence of c by VALUED_0:def_17;
then A73: rng c2 = {x0} by A4, VALUED_0:26;
then A74: lim ((h2 ") (#) ((f1 /* (h2 + c2)) - (f1 /* c2))) = diff (f1,x0) by A2, A72, Th12;
diff (f1,x0) = diff (f1,x0) ;
then A75: (h2 ") (#) ((f1 /* (h2 + c2)) - (f1 /* c2)) is convergent by A2, A73, A72, Th12;
diff (f2,(f1 . x0)) = diff (f2,(f1 . x0)) ;
then A76: (d1 ") (#) ((f2 /* (d1 + (a * I2))) - (f2 /* (a * I2))) is convergent by A3, A67, A71, Th12;
then A77: ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) * I2 is convergent by A70, A75, SEQ_2:14;
lim ((d1 ") (#) ((f2 /* (d1 + (a * I2))) - (f2 /* (a * I2)))) = diff (f2,(f1 . x0)) by A3, A67, A71, Th12;
then A78: lim (((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) * I2) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) by A70, A75, A74, A76, SEQ_2:15;
A79: now__::_thesis:_for_g_being_real_number_st_0_<_g_holds_
ex_n_being_Element_of_NAT_st_
for_m_being_Element_of_NAT_st_n_<=_m_holds_
abs_((((h_")_(#)_(((f2_*_f1)_/*_(h_+_c))_-_((f2_*_f1)_/*_c)))_._m)_-_((diff_(f2,(f1_._x0)))_*_(diff_(f1,x0))))_<_g
set DF = (diff (f2,(f1 . x0))) * (diff (f1,x0));
let g be real number ; ::_thesis: ( 0 < g implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g )
assume A80: 0 < g ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g
consider k being Element of NAT such that
A81: for m being Element of NAT st k <= m holds
abs (((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) * I2) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g by A77, A78, A80, SEQ_2:def_7;
take n = I2 . k; ::_thesis: for m being Element of NAT st n <= m holds
abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g
let m be Element of NAT ; ::_thesis: ( n <= m implies abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g )
assume A82: n <= m ; ::_thesis: abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g
now__::_thesis:_abs_((((h_")_(#)_(((f2_*_f1)_/*_(h_+_c))_-_((f2_*_f1)_/*_c)))_._m)_-_((diff_(f2,(f1_._x0)))_*_(diff_(f1,x0))))_<_g
percases ( (f1 /* (h + c)) . m = f1 . x0 or (f1 /* (h + c)) . m <> f1 . x0 ) ;
supposeA83: (f1 /* (h + c)) . m = f1 . x0 ; ::_thesis: abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g
abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) = abs (((h ") . m) * ((((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) . m)) by A54, SEQ_1:8
.= abs (((h ") . m) * ((((f2 * f1) /* (h + c)) . m) - (((f2 * f1) /* c) . m))) by RFUNCT_2:1
.= abs (((h ") . m) * ((((f2 * f1) /* (h + c)) . m) - ((f2 /* (f1 /* c)) . m))) by A8, VALUED_0:31
.= abs (((h ") . m) * ((((f2 * f1) /* (h + c)) . m) - (f2 . ((f1 /* c) . m)))) by A24, FUNCT_2:108
.= abs (((h ") . m) * ((((f2 * f1) /* (h + c)) . m) - (f2 . (f1 . (c . m))))) by A18, FUNCT_2:108
.= abs (((h ") . m) * ((((f2 * f1) /* (h + c)) . m) - (f2 . (f1 . x0)))) by A7
.= abs (((h ") . m) * (((f2 /* (f1 /* (h + c))) . m) - (f2 . (f1 . x0)))) by A5, VALUED_0:31
.= abs (((h ") . m) * ((f2 . (f1 . x0)) - (f2 . (f1 . x0)))) by A17, A83, FUNCT_2:108
.= 0 by ABSVALUE:2 ;
hence abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g by A80; ::_thesis: verum
end;
suppose (f1 /* (h + c)) . m <> f1 . x0 ; ::_thesis: abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g
then for r1 being Real st (f1 /* (h + c)) . m = r1 holds
r1 <> f1 . x0 ;
then consider l being Element of NAT such that
A84: m = I2 . l by A62;
dom I2 = NAT by FUNCT_2:def_1;
then l >= k by A82, A84, VALUED_0:def_13;
then abs (((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) * I2) . l) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g by A81;
hence abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g by A84, FUNCT_2:15; ::_thesis: verum
end;
end;
end;
hence abs ((((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) . m) - ((diff (f2,(f1 . x0))) * (diff (f1,x0)))) < g ; ::_thesis: verum
end;
hence (h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) is convergent by SEQ_2:def_6; ::_thesis: lim ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) = (diff (f2,(f1 . x0))) * (diff (f1,x0))
hence lim ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) by A79, SEQ_2:def_7; ::_thesis: verum
end;
end;
end;
hence ( (h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) is convergent & lim ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) ) ; ::_thesis: verum
end;
end;
end;
hence ( (h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c)) is convergent & lim ((h ") (#) (((f2 * f1) /* (h + c)) - ((f2 * f1) /* c))) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) ) ; ::_thesis: verum
end;
hence ( f2 * f1 is_differentiable_in x0 & diff ((f2 * f1),x0) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) ) by A1, Th12; ::_thesis: verum
end;
theorem Th13: :: FDIFF_2:13
for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_differentiable_in x0 & f2 is_differentiable_in f1 . x0 holds
( f2 * f1 is_differentiable_in x0 & diff ((f2 * f1),x0) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) )
proof
let x0 be Real; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_differentiable_in x0 & f2 is_differentiable_in f1 . x0 holds
( f2 * f1 is_differentiable_in x0 & diff ((f2 * f1),x0) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) )
let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_differentiable_in x0 & f2 is_differentiable_in f1 . x0 implies ( f2 * f1 is_differentiable_in x0 & diff ((f2 * f1),x0) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) ) )
assume that
A1: f1 is_differentiable_in x0 and
A2: f2 is_differentiable_in f1 . x0 ; ::_thesis: ( f2 * f1 is_differentiable_in x0 & diff ((f2 * f1),x0) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) )
consider N2 being Neighbourhood of f1 . x0 such that
A3: N2 c= dom f2 by A2, Th11;
f1 is_continuous_in x0 by A1, FDIFF_1:24;
then consider N3 being Neighbourhood of x0 such that
A4: f1 .: N3 c= N2 by FCONT_1:5;
consider N1 being Neighbourhood of x0 such that
A5: N1 c= dom f1 by A1, Th11;
consider N being Neighbourhood of x0 such that
A6: N c= N1 and
A7: N c= N3 by RCOMP_1:17;
N c= dom (f2 * f1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in N or x in dom (f2 * f1) )
assume A8: x in N ; ::_thesis: x in dom (f2 * f1)
then reconsider x9 = x as Real ;
A9: x in N1 by A6, A8;
then f1 . x9 in f1 .: N3 by A5, A7, A8, FUNCT_1:def_6;
then f1 . x9 in N2 by A4;
hence x in dom (f2 * f1) by A5, A3, A9, FUNCT_1:11; ::_thesis: verum
end;
hence ( f2 * f1 is_differentiable_in x0 & diff ((f2 * f1),x0) = (diff (f2,(f1 . x0))) * (diff (f1,x0)) ) by A1, A2, Lm2; ::_thesis: verum
end;
theorem Th14: :: FDIFF_2:14
for x0 being Real
for f2, f1 being PartFunc of REAL,REAL st f2 . x0 <> 0 & f1 is_differentiable_in x0 & f2 is_differentiable_in x0 holds
( f1 / f2 is_differentiable_in x0 & diff ((f1 / f2),x0) = (((diff (f1,x0)) * (f2 . x0)) - ((diff (f2,x0)) * (f1 . x0))) / ((f2 . x0) ^2) )
proof
let x0 be Real; ::_thesis: for f2, f1 being PartFunc of REAL,REAL st f2 . x0 <> 0 & f1 is_differentiable_in x0 & f2 is_differentiable_in x0 holds
( f1 / f2 is_differentiable_in x0 & diff ((f1 / f2),x0) = (((diff (f1,x0)) * (f2 . x0)) - ((diff (f2,x0)) * (f1 . x0))) / ((f2 . x0) ^2) )
let f2, f1 be PartFunc of REAL,REAL; ::_thesis: ( f2 . x0 <> 0 & f1 is_differentiable_in x0 & f2 is_differentiable_in x0 implies ( f1 / f2 is_differentiable_in x0 & diff ((f1 / f2),x0) = (((diff (f1,x0)) * (f2 . x0)) - ((diff (f2,x0)) * (f1 . x0))) / ((f2 . x0) ^2) ) )
assume that
A1: f2 . x0 <> 0 and
A2: f1 is_differentiable_in x0 and
A3: f2 is_differentiable_in x0 ; ::_thesis: ( f1 / f2 is_differentiable_in x0 & diff ((f1 / f2),x0) = (((diff (f1,x0)) * (f2 . x0)) - ((diff (f2,x0)) * (f1 . x0))) / ((f2 . x0) ^2) )
consider N1 being Neighbourhood of x0 such that
A4: N1 c= dom f1 by A2, Th11;
ex N2 being Neighbourhood of x0 st N2 c= dom f2 by A3, Th11;
then consider N3 being Neighbourhood of x0 such that
A5: N3 c= dom f2 and
A6: for g being Real st g in N3 holds
f2 . g <> 0 by A1, A3, FCONT_3:7, FDIFF_1:24;
consider N being Neighbourhood of x0 such that
A7: N c= N1 and
A8: N c= N3 by RCOMP_1:17;
A9: N c= (dom f2) \ (f2 " {0})
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in N or x in (dom f2) \ (f2 " {0}) )
assume A10: x in N ; ::_thesis: x in (dom f2) \ (f2 " {0})
then reconsider x9 = x as Real ;
f2 . x9 <> 0 by A6, A8, A10;
then not f2 . x9 in {0} by TARSKI:def_1;
then A11: not x9 in f2 " {0} by FUNCT_1:def_7;
x9 in N3 by A8, A10;
hence x in (dom f2) \ (f2 " {0}) by A5, A11, XBOOLE_0:def_5; ::_thesis: verum
end;
A12: f2 is_continuous_in x0 by A3, FDIFF_1:24;
N c= dom f1 by A4, A7, XBOOLE_1:1;
then A13: N c= (dom f1) /\ ((dom f2) \ (f2 " {0})) by A9, XBOOLE_1:19;
A14: for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom (f1 / f2) holds
( (h ") (#) (((f1 / f2) /* (h + c)) - ((f1 / f2) /* c)) is convergent & lim ((h ") (#) (((f1 / f2) /* (h + c)) - ((f1 / f2) /* c))) = (((diff (f1,x0)) * (f2 . x0)) - ((diff (f2,x0)) * (f1 . x0))) / ((f2 . x0) ^2) )
proof
dom (f2 ^) = (dom f2) \ (f2 " {0}) by RFUNCT_1:def_2;
then A15: dom (f2 ^) c= dom f2 by XBOOLE_1:36;
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom (f1 / f2) holds
( (h ") (#) (((f1 / f2) /* (h + c)) - ((f1 / f2) /* c)) is convergent & lim ((h ") (#) (((f1 / f2) /* (h + c)) - ((f1 / f2) /* c))) = (((diff (f1,x0)) * (f2 . x0)) - ((diff (f2,x0)) * (f1 . x0))) / ((f2 . x0) ^2) )
let c be V8() Real_Sequence; ::_thesis: ( rng c = {x0} & rng (h + c) c= dom (f1 / f2) implies ( (h ") (#) (((f1 / f2) /* (h + c)) - ((f1 / f2) /* c)) is convergent & lim ((h ") (#) (((f1 / f2) /* (h + c)) - ((f1 / f2) /* c))) = (((diff (f1,x0)) * (f2 . x0)) - ((diff (f2,x0)) * (f1 . x0))) / ((f2 . x0) ^2) ) )
assume that
A16: rng c = {x0} and
A17: rng (h + c) c= dom (f1 / f2) ; ::_thesis: ( (h ") (#) (((f1 / f2) /* (h + c)) - ((f1 / f2) /* c)) is convergent & lim ((h ") (#) (((f1 / f2) /* (h + c)) - ((f1 / f2) /* c))) = (((diff (f1,x0)) * (f2 . x0)) - ((diff (f2,x0)) * (f1 . x0))) / ((f2 . x0) ^2) )
A18: rng (h + c) c= (dom f1) /\ ((dom f2) \ (f2 " {0})) by A17, RFUNCT_1:def_1;
(dom f1) /\ ((dom f2) \ (f2 " {0})) c= (dom f2) \ (f2 " {0}) by XBOOLE_1:17;
then A19: rng (h + c) c= (dom f2) \ (f2 " {0}) by A18, XBOOLE_1:1;
then A20: rng (h + c) c= dom (f2 ^) by RFUNCT_1:def_2;
then A21: f2 /* (h + c) is non-zero by RFUNCT_2:11;
A22: lim c = x0 by A16, Th4;
set u2 = f2 /* c;
set u1 = f1 /* c;
set v2 = f2 /* (h + c);
set h2 = (h ") (#) ((f2 /* (h + c)) - (f2 /* c));
set v1 = f1 /* (h + c);
set h1 = (h ") (#) ((f1 /* (h + c)) - (f1 /* c));
A23: f1 is_continuous_in x0 by A2, FDIFF_1:24;
A24: rng c c= (dom f1) /\ (dom (f2 ^))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng c or x in (dom f1) /\ (dom (f2 ^)) )
assume x in rng c ; ::_thesis: x in (dom f1) /\ (dom (f2 ^))
then A25: x = x0 by A16, TARSKI:def_1;
x0 in N by RCOMP_1:16;
then x in (dom f1) /\ ((dom f2) \ (f2 " {0})) by A13, A25;
hence x in (dom f1) /\ (dom (f2 ^)) by RFUNCT_1:def_2; ::_thesis: verum
end;
A26: (dom f1) /\ (dom (f2 ^)) c= dom (f2 ^) by XBOOLE_1:17;
then A27: f2 /* c is non-zero by A24, RFUNCT_2:11, XBOOLE_1:1;
(dom f1) /\ ((dom f2) \ (f2 " {0})) c= dom f1 by XBOOLE_1:17;
then A28: rng (h + c) c= dom f1 by A18, XBOOLE_1:1;
then A29: rng (h + c) c= (dom f1) /\ (dom (f2 ^)) by A20, XBOOLE_1:19;
A30: (h ") (#) (((f1 / f2) /* (h + c)) - ((f1 / f2) /* c)) = (h ") (#) (((f1 (#) (f2 ^)) /* (h + c)) - ((f1 / f2) /* c)) by RFUNCT_1:31
.= (h ") (#) (((f1 (#) (f2 ^)) /* (h + c)) - ((f1 (#) (f2 ^)) /* c)) by RFUNCT_1:31
.= (h ") (#) (((f1 /* (h + c)) (#) ((f2 ^) /* (h + c))) - ((f1 (#) (f2 ^)) /* c)) by A29, RFUNCT_2:8
.= (h ") (#) (((f1 /* (h + c)) /" (f2 /* (h + c))) - ((f1 (#) (f2 ^)) /* c)) by A20, RFUNCT_2:12
.= (h ") (#) (((f1 /* (h + c)) /" (f2 /* (h + c))) - ((f1 /* c) (#) ((f2 ^) /* c))) by A24, RFUNCT_2:8
.= (h ") (#) (((f1 /* (h + c)) /" (f2 /* (h + c))) - ((f1 /* c) /" (f2 /* c))) by A24, A26, RFUNCT_2:12, XBOOLE_1:1
.= (h ") (#) ((((f1 /* (h + c)) (#) (f2 /* c)) - ((f1 /* c) (#) (f2 /* (h + c)))) /" ((f2 /* (h + c)) (#) (f2 /* c))) by A21, A27, SEQ_1:50
.= ((h ") (#) (((f1 /* (h + c)) (#) (f2 /* c)) - ((f1 /* c) (#) (f2 /* (h + c))))) /" ((f2 /* (h + c)) (#) (f2 /* c)) by SEQ_1:41 ;
rng c c= dom (f2 ^) by A24, A26, XBOOLE_1:1;
then A31: rng c c= dom f2 by A15, XBOOLE_1:1;
then A32: f2 /* c is convergent by A12, A22, FCONT_1:def_1;
(dom f1) /\ (dom (f2 ^)) c= dom f1 by XBOOLE_1:17;
then A33: rng c c= dom f1 by A24, XBOOLE_1:1;
then A34: lim (f1 /* c) = f1 . x0 by A22, A23, FCONT_1:def_1;
(dom f2) \ (f2 " {0}) c= dom f2 by XBOOLE_1:36;
then A35: rng (h + c) c= dom f2 by A19, XBOOLE_1:1;
diff (f2,x0) = diff (f2,x0) ;
then A36: (h ") (#) ((f2 /* (h + c)) - (f2 /* c)) is convergent by A3, A16, A35, Th12;
A37: f1 /* c is convergent by A33, A22, A23, FCONT_1:def_1;
then A38: (f1 /* c) (#) ((h ") (#) ((f2 /* (h + c)) - (f2 /* c))) is convergent by A36, SEQ_2:14;
lim ((h ") (#) ((f2 /* (h + c)) - (f2 /* c))) = diff (f2,x0) by A3, A16, A35, Th12;
then A39: lim ((f1 /* c) (#) ((h ") (#) ((f2 /* (h + c)) - (f2 /* c)))) = (diff (f2,x0)) * (f1 . x0) by A36, A37, A34, SEQ_2:15;
set w1 = (f2 /* (h + c)) (#) (f2 /* c);
A40: lim (h + c) = x0 by A16, Th4;
A41: h + c is convergent by A16, Th4;
then A42: f2 /* (h + c) is convergent by A12, A35, A40, FCONT_1:def_1;
then A43: (f2 /* (h + c)) (#) (f2 /* c) is convergent by A32, SEQ_2:14;
f2 /* (h + c) is non-zero by A20, RFUNCT_2:11;
then A44: (f2 /* (h + c)) (#) (f2 /* c) is non-zero by A27, SEQ_1:35;
A45: lim (f2 /* c) = f2 . x0 by A12, A31, A22, FCONT_1:def_1;
diff (f1,x0) = diff (f1,x0) ;
then A46: (h ") (#) ((f1 /* (h + c)) - (f1 /* c)) is convergent by A2, A16, A28, Th12;
then A47: ((h ") (#) ((f1 /* (h + c)) - (f1 /* c))) (#) (f2 /* c) is convergent by A32, SEQ_2:14;
lim (f2 /* (h + c)) = f2 . x0 by A12, A35, A41, A40, FCONT_1:def_1;
then A48: lim ((f2 /* (h + c)) (#) (f2 /* c)) = (f2 . x0) ^2 by A42, A32, A45, SEQ_2:15;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((h_")_(#)_(((f1_/*_(h_+_c))_(#)_(f2_/*_c))_-_((f1_/*_c)_(#)_(f2_/*_(h_+_c)))))_._n_=_((((h_")_(#)_((f1_/*_(h_+_c))_-_(f1_/*_c)))_(#)_(f2_/*_c))_-_((f1_/*_c)_(#)_((h_")_(#)_((f2_/*_(h_+_c))_-_(f2_/*_c)))))_._n
let n be Element of NAT ; ::_thesis: ((h ") (#) (((f1 /* (h + c)) (#) (f2 /* c)) - ((f1 /* c) (#) (f2 /* (h + c))))) . n = ((((h ") (#) ((f1 /* (h + c)) - (f1 /* c))) (#) (f2 /* c)) - ((f1 /* c) (#) ((h ") (#) ((f2 /* (h + c)) - (f2 /* c))))) . n
thus ((h ") (#) (((f1 /* (h + c)) (#) (f2 /* c)) - ((f1 /* c) (#) (f2 /* (h + c))))) . n = ((h ") . n) * ((((f1 /* (h + c)) (#) (f2 /* c)) - ((f1 /* c) (#) (f2 /* (h + c)))) . n) by SEQ_1:8
.= ((h ") . n) * ((((f1 /* (h + c)) (#) (f2 /* c)) . n) - (((f1 /* c) (#) (f2 /* (h + c))) . n)) by RFUNCT_2:1
.= ((h ") . n) * ((((f1 /* (h + c)) . n) * ((f2 /* c) . n)) - (((f1 /* c) (#) (f2 /* (h + c))) . n)) by SEQ_1:8
.= ((h ") . n) * ((((((f1 /* (h + c)) . n) - ((f1 /* c) . n)) * ((f2 /* c) . n)) + (((f1 /* c) . n) * ((f2 /* c) . n))) - (((f1 /* c) . n) * ((f2 /* (h + c)) . n))) by SEQ_1:8
.= ((((h ") . n) * (((f1 /* (h + c)) . n) - ((f1 /* c) . n))) * ((f2 /* c) . n)) - (((f1 /* c) . n) * (((h ") . n) * (((f2 /* (h + c)) . n) - ((f2 /* c) . n))))
.= ((((h ") . n) * (((f1 /* (h + c)) - (f1 /* c)) . n)) * ((f2 /* c) . n)) - (((f1 /* c) . n) * (((h ") . n) * (((f2 /* (h + c)) . n) - ((f2 /* c) . n)))) by RFUNCT_2:1
.= ((((h ") . n) * (((f1 /* (h + c)) - (f1 /* c)) . n)) * ((f2 /* c) . n)) - (((f1 /* c) . n) * (((h ") . n) * (((f2 /* (h + c)) - (f2 /* c)) . n))) by RFUNCT_2:1
.= ((((h ") (#) ((f1 /* (h + c)) - (f1 /* c))) . n) * ((f2 /* c) . n)) - (((f1 /* c) . n) * (((h ") . n) * (((f2 /* (h + c)) - (f2 /* c)) . n))) by SEQ_1:8
.= ((((h ") (#) ((f1 /* (h + c)) - (f1 /* c))) . n) * ((f2 /* c) . n)) - (((f1 /* c) . n) * (((h ") (#) ((f2 /* (h + c)) - (f2 /* c))) . n)) by SEQ_1:8
.= ((((h ") (#) ((f1 /* (h + c)) - (f1 /* c))) (#) (f2 /* c)) . n) - (((f1 /* c) . n) * (((h ") (#) ((f2 /* (h + c)) - (f2 /* c))) . n)) by SEQ_1:8
.= ((((h ") (#) ((f1 /* (h + c)) - (f1 /* c))) (#) (f2 /* c)) . n) - (((f1 /* c) (#) ((h ") (#) ((f2 /* (h + c)) - (f2 /* c)))) . n) by SEQ_1:8
.= ((((h ") (#) ((f1 /* (h + c)) - (f1 /* c))) (#) (f2 /* c)) - ((f1 /* c) (#) ((h ") (#) ((f2 /* (h + c)) - (f2 /* c))))) . n by RFUNCT_2:1 ; ::_thesis: verum
end;
then A49: (h ") (#) (((f1 /* (h + c)) (#) (f2 /* c)) - ((f1 /* c) (#) (f2 /* (h + c)))) = (((h ") (#) ((f1 /* (h + c)) - (f1 /* c))) (#) (f2 /* c)) - ((f1 /* c) (#) ((h ") (#) ((f2 /* (h + c)) - (f2 /* c)))) by FUNCT_2:63;
then A50: (h ") (#) (((f1 /* (h + c)) (#) (f2 /* c)) - ((f1 /* c) (#) (f2 /* (h + c)))) is convergent by A47, A38, SEQ_2:11;
hence (h ") (#) (((f1 / f2) /* (h + c)) - ((f1 / f2) /* c)) is convergent by A1, A30, A43, A48, A44, SEQ_2:23; ::_thesis: lim ((h ") (#) (((f1 / f2) /* (h + c)) - ((f1 / f2) /* c))) = (((diff (f1,x0)) * (f2 . x0)) - ((diff (f2,x0)) * (f1 . x0))) / ((f2 . x0) ^2)
lim ((h ") (#) ((f1 /* (h + c)) - (f1 /* c))) = diff (f1,x0) by A2, A16, A28, Th12;
then lim (((h ") (#) ((f1 /* (h + c)) - (f1 /* c))) (#) (f2 /* c)) = (diff (f1,x0)) * (f2 . x0) by A32, A45, A46, SEQ_2:15;
then lim ((h ") (#) (((f1 /* (h + c)) (#) (f2 /* c)) - ((f1 /* c) (#) (f2 /* (h + c))))) = ((diff (f1,x0)) * (f2 . x0)) - ((diff (f2,x0)) * (f1 . x0)) by A49, A47, A38, A39, SEQ_2:12;
hence lim ((h ") (#) (((f1 / f2) /* (h + c)) - ((f1 / f2) /* c))) = (((diff (f1,x0)) * (f2 . x0)) - ((diff (f2,x0)) * (f1 . x0))) / ((f2 . x0) ^2) by A1, A30, A43, A48, A50, A44, SEQ_2:24; ::_thesis: verum
end;
N c= dom (f1 / f2) by A13, RFUNCT_1:def_1;
hence ( f1 / f2 is_differentiable_in x0 & diff ((f1 / f2),x0) = (((diff (f1,x0)) * (f2 . x0)) - ((diff (f2,x0)) * (f1 . x0))) / ((f2 . x0) ^2) ) by A14, Th12; ::_thesis: verum
end;
theorem Th15: :: FDIFF_2:15
for x0 being Real
for f being PartFunc of REAL,REAL st f . x0 <> 0 & f is_differentiable_in x0 holds
( f ^ is_differentiable_in x0 & diff ((f ^),x0) = - ((diff (f,x0)) / ((f . x0) ^2)) )
proof
let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f . x0 <> 0 & f is_differentiable_in x0 holds
( f ^ is_differentiable_in x0 & diff ((f ^),x0) = - ((diff (f,x0)) / ((f . x0) ^2)) )
let f be PartFunc of REAL,REAL; ::_thesis: ( f . x0 <> 0 & f is_differentiable_in x0 implies ( f ^ is_differentiable_in x0 & diff ((f ^),x0) = - ((diff (f,x0)) / ((f . x0) ^2)) ) )
reconsider f1 = (dom f) --> 1 as PartFunc of (dom f),REAL by FUNCOP_1:45;
dom f1 c= REAL by RELAT_1:def_18;
then reconsider f1 = f1 as PartFunc of REAL,REAL by RELSET_1:5;
assume that
A1: f . x0 <> 0 and
A2: f is_differentiable_in x0 ; ::_thesis: ( f ^ is_differentiable_in x0 & diff ((f ^),x0) = - ((diff (f,x0)) / ((f . x0) ^2)) )
consider N being Neighbourhood of x0 such that
A3: N c= dom f by A2, Th11;
A4: x0 in N by RCOMP_1:16;
A5: dom f1 = dom f by FUNCOP_1:13;
A6: rng f1 = {1}
proof
thus rng f1 c= {1} by FUNCOP_1:13; :: according to XBOOLE_0:def_10 ::_thesis: {1} c= rng f1
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {1} or x in rng f1 )
A7: x0 in N by RCOMP_1:16;
assume x in {1} ; ::_thesis: x in rng f1
then x = 1 by TARSKI:def_1;
then f1 . x0 = x by A3, A7, FUNCOP_1:7;
hence x in rng f1 by A5, A3, A7, FUNCT_1:def_3; ::_thesis: verum
end;
then A8: f1 is_differentiable_on N by A5, A3, FDIFF_1:11;
then A9: f1 is_differentiable_in x0 by A4, FDIFF_1:9;
0 = (f1 `| N) . x0 by A5, A3, A6, FDIFF_1:11, RCOMP_1:16
.= diff (f1,x0) by A8, A4, FDIFF_1:def_7 ;
then diff ((f1 / f),x0) = ((0 * (f . x0)) - ((diff (f,x0)) * (f1 . x0))) / ((f . x0) ^2) by A1, A2, A9, Th14;
then A10: diff ((f1 / f),x0) = (- ((diff (f,x0)) * (f1 . x0))) / ((f . x0) ^2)
.= (- ((diff (f,x0)) * 1)) / ((f . x0) ^2) by A3, A4, FUNCOP_1:7
.= - ((diff (f,x0)) / ((f . x0) ^2)) by XCMPLX_1:187 ;
A11: dom (f1 / f) = (dom f1) /\ ((dom f) \ (f " {0})) by RFUNCT_1:def_1
.= (dom f) \ (f " {0}) by A5, XBOOLE_1:28, XBOOLE_1:36
.= dom (f ^) by RFUNCT_1:def_2 ;
A12: (dom f) \ (f " {0}) c= dom f1 by A5, XBOOLE_1:36;
A13: now__::_thesis:_for_r_being_Real_st_r_in_dom_(f1_/_f)_holds_
(f1_/_f)_._r_=_(f_^)_._r
A14: (dom f) \ (f " {0}) = dom (f ^) by RFUNCT_1:def_2;
let r be Real; ::_thesis: ( r in dom (f1 / f) implies (f1 / f) . r = (f ^) . r )
assume A15: r in dom (f1 / f) ; ::_thesis: (f1 / f) . r = (f ^) . r
thus (f1 / f) . r = (f1 . r) * ((f . r) ") by A15, RFUNCT_1:def_1
.= 1 * ((f . r) ") by A5, A12, A11, A15, A14, FUNCOP_1:7
.= (f ^) . r by A11, A15, RFUNCT_1:def_2 ; ::_thesis: verum
end;
f1 / f is_differentiable_in x0 by A1, A2, A9, Th14;
hence ( f ^ is_differentiable_in x0 & diff ((f ^),x0) = - ((diff (f,x0)) / ((f . x0) ^2)) ) by A10, A11, A13, PARTFUN1:5; ::_thesis: verum
end;
theorem :: FDIFF_2:16
for A being open Subset of REAL
for f being PartFunc of REAL,REAL st f is_differentiable_on A holds
( f | A is_differentiable_on A & f `| A = (f | A) `| A )
proof
let A be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st f is_differentiable_on A holds
( f | A is_differentiable_on A & f `| A = (f | A) `| A )
let f be PartFunc of REAL,REAL; ::_thesis: ( f is_differentiable_on A implies ( f | A is_differentiable_on A & f `| A = (f | A) `| A ) )
assume A1: f is_differentiable_on A ; ::_thesis: ( f | A is_differentiable_on A & f `| A = (f | A) `| A )
then A c= dom f by FDIFF_1:def_6;
then A c= (dom f) /\ A by XBOOLE_1:19;
then A2: A c= dom (f | A) by RELAT_1:61;
now__::_thesis:_for_x0_being_Real_st_x0_in_A_holds_
(f_|_A)_|_A_is_differentiable_in_x0
let x0 be Real; ::_thesis: ( x0 in A implies (f | A) | A is_differentiable_in x0 )
assume x0 in A ; ::_thesis: (f | A) | A is_differentiable_in x0
then f | A is_differentiable_in x0 by A1, FDIFF_1:def_6;
hence (f | A) | A is_differentiable_in x0 by RELAT_1:72; ::_thesis: verum
end;
hence A3: f | A is_differentiable_on A by A2, FDIFF_1:def_6; ::_thesis: f `| A = (f | A) `| A
then A4: dom ((f | A) `| A) = A by FDIFF_1:def_7;
A5: now__::_thesis:_for_x0_being_Real_st_x0_in_A_holds_
(f_`|_A)_._x0_=_((f_|_A)_`|_A)_._x0
let x0 be Real; ::_thesis: ( x0 in A implies (f `| A) . x0 = ((f | A) `| A) . x0 )
assume A6: x0 in A ; ::_thesis: (f `| A) . x0 = ((f | A) `| A) . x0
then consider N2 being Neighbourhood of x0 such that
A7: N2 c= A by RCOMP_1:18;
A8: f | A is_differentiable_in x0 by A1, A6, FDIFF_1:def_6;
A9: f is_differentiable_in x0 by A1, A6, FDIFF_1:9;
then consider N1 being Neighbourhood of x0 such that
A10: N1 c= dom f and
A11: ex L being LinearFunc ex R being RestFunc st
for r being Real st r in N1 holds
(f . r) - (f . x0) = (L . (r - x0)) + (R . (r - x0)) by FDIFF_1:def_4;
consider L being LinearFunc, R being RestFunc such that
A12: for r being Real st r in N1 holds
(f . r) - (f . x0) = (L . (r - x0)) + (R . (r - x0)) by A11;
consider N being Neighbourhood of x0 such that
A13: N c= N1 and
A14: N c= N2 by RCOMP_1:17;
A15: N c= A by A7, A14, XBOOLE_1:1;
then A16: N c= dom (f | A) by A2, XBOOLE_1:1;
A17: now__::_thesis:_for_r_being_Real_st_r_in_N_holds_
((f_|_A)_._r)_-_((f_|_A)_._x0)_=_(L_._(r_-_x0))_+_(R_._(r_-_x0))
let r be Real; ::_thesis: ( r in N implies ((f | A) . r) - ((f | A) . x0) = (L . (r - x0)) + (R . (r - x0)) )
assume A18: r in N ; ::_thesis: ((f | A) . r) - ((f | A) . x0) = (L . (r - x0)) + (R . (r - x0))
then A19: r in A by A15;
thus ((f | A) . r) - ((f | A) . x0) = ((f | A) . r) - (f . x0) by A2, A6, FUNCT_1:47
.= (f . r) - (f . x0) by A2, A19, FUNCT_1:47
.= (L . (r - x0)) + (R . (r - x0)) by A12, A13, A18 ; ::_thesis: verum
end;
thus (f `| A) . x0 = diff (f,x0) by A1, A6, FDIFF_1:def_7
.= L . 1 by A9, A10, A12, FDIFF_1:def_5
.= diff ((f | A),x0) by A8, A16, A17, FDIFF_1:def_5
.= ((f | A) `| A) . x0 by A3, A6, FDIFF_1:def_7 ; ::_thesis: verum
end;
dom (f `| A) = A by A1, FDIFF_1:def_7;
hence f `| A = (f | A) `| A by A4, A5, PARTFUN1:5; ::_thesis: verum
end;
theorem :: FDIFF_2:17
for A being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st f1 is_differentiable_on A & f2 is_differentiable_on A holds
( f1 + f2 is_differentiable_on A & (f1 + f2) `| A = (f1 `| A) + (f2 `| A) )
proof
let A be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_differentiable_on A & f2 is_differentiable_on A holds
( f1 + f2 is_differentiable_on A & (f1 + f2) `| A = (f1 `| A) + (f2 `| A) )
let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_differentiable_on A & f2 is_differentiable_on A implies ( f1 + f2 is_differentiable_on A & (f1 + f2) `| A = (f1 `| A) + (f2 `| A) ) )
assume that
A1: f1 is_differentiable_on A and
A2: f2 is_differentiable_on A ; ::_thesis: ( f1 + f2 is_differentiable_on A & (f1 + f2) `| A = (f1 `| A) + (f2 `| A) )
A3: A c= dom f2 by A2, FDIFF_1:def_6;
A c= dom f1 by A1, FDIFF_1:def_6;
then A c= (dom f1) /\ (dom f2) by A3, XBOOLE_1:19;
then A4: A c= dom (f1 + f2) by VALUED_1:def_1;
then f1 + f2 is_differentiable_on A by A1, A2, FDIFF_1:18;
then A5: dom ((f1 + f2) `| A) = A by FDIFF_1:def_7;
A6: dom (f2 `| A) = A by A2, FDIFF_1:def_7;
dom (f1 `| A) = A by A1, FDIFF_1:def_7;
then (dom (f1 `| A)) /\ (dom (f2 `| A)) = A by A6;
then A7: dom ((f1 `| A) + (f2 `| A)) = A by VALUED_1:def_1;
now__::_thesis:_for_x0_being_Real_st_x0_in_A_holds_
((f1_+_f2)_`|_A)_._x0_=_((f1_`|_A)_+_(f2_`|_A))_._x0
let x0 be Real; ::_thesis: ( x0 in A implies ((f1 + f2) `| A) . x0 = ((f1 `| A) + (f2 `| A)) . x0 )
assume A8: x0 in A ; ::_thesis: ((f1 + f2) `| A) . x0 = ((f1 `| A) + (f2 `| A)) . x0
hence ((f1 + f2) `| A) . x0 = (diff (f1,x0)) + (diff (f2,x0)) by A1, A2, A4, FDIFF_1:18
.= ((f1 `| A) . x0) + (diff (f2,x0)) by A1, A8, FDIFF_1:def_7
.= ((f1 `| A) . x0) + ((f2 `| A) . x0) by A2, A8, FDIFF_1:def_7
.= ((f1 `| A) + (f2 `| A)) . x0 by A7, A8, VALUED_1:def_1 ;
::_thesis: verum
end;
hence ( f1 + f2 is_differentiable_on A & (f1 + f2) `| A = (f1 `| A) + (f2 `| A) ) by A1, A2, A4, A5, A7, FDIFF_1:18, PARTFUN1:5; ::_thesis: verum
end;
theorem :: FDIFF_2:18
for A being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st f1 is_differentiable_on A & f2 is_differentiable_on A holds
( f1 - f2 is_differentiable_on A & (f1 - f2) `| A = (f1 `| A) - (f2 `| A) )
proof
let A be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_differentiable_on A & f2 is_differentiable_on A holds
( f1 - f2 is_differentiable_on A & (f1 - f2) `| A = (f1 `| A) - (f2 `| A) )
let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_differentiable_on A & f2 is_differentiable_on A implies ( f1 - f2 is_differentiable_on A & (f1 - f2) `| A = (f1 `| A) - (f2 `| A) ) )
assume that
A1: f1 is_differentiable_on A and
A2: f2 is_differentiable_on A ; ::_thesis: ( f1 - f2 is_differentiable_on A & (f1 - f2) `| A = (f1 `| A) - (f2 `| A) )
A3: A c= dom f2 by A2, FDIFF_1:def_6;
A c= dom f1 by A1, FDIFF_1:def_6;
then A c= (dom f1) /\ (dom f2) by A3, XBOOLE_1:19;
then A4: A c= dom (f1 - f2) by VALUED_1:12;
then f1 - f2 is_differentiable_on A by A1, A2, FDIFF_1:19;
then A5: dom ((f1 - f2) `| A) = A by FDIFF_1:def_7;
A6: dom (f2 `| A) = A by A2, FDIFF_1:def_7;
dom (f1 `| A) = A by A1, FDIFF_1:def_7;
then (dom (f1 `| A)) /\ (dom (f2 `| A)) = A by A6;
then A7: dom ((f1 `| A) - (f2 `| A)) = A by VALUED_1:12;
now__::_thesis:_for_x0_being_Real_st_x0_in_A_holds_
((f1_-_f2)_`|_A)_._x0_=_((f1_`|_A)_-_(f2_`|_A))_._x0
let x0 be Real; ::_thesis: ( x0 in A implies ((f1 - f2) `| A) . x0 = ((f1 `| A) - (f2 `| A)) . x0 )
assume A8: x0 in A ; ::_thesis: ((f1 - f2) `| A) . x0 = ((f1 `| A) - (f2 `| A)) . x0
hence ((f1 - f2) `| A) . x0 = (diff (f1,x0)) - (diff (f2,x0)) by A1, A2, A4, FDIFF_1:19
.= ((f1 `| A) . x0) - (diff (f2,x0)) by A1, A8, FDIFF_1:def_7
.= ((f1 `| A) . x0) - ((f2 `| A) . x0) by A2, A8, FDIFF_1:def_7
.= ((f1 `| A) - (f2 `| A)) . x0 by A7, A8, VALUED_1:13 ;
::_thesis: verum
end;
hence ( f1 - f2 is_differentiable_on A & (f1 - f2) `| A = (f1 `| A) - (f2 `| A) ) by A1, A2, A4, A5, A7, FDIFF_1:19, PARTFUN1:5; ::_thesis: verum
end;
theorem :: FDIFF_2:19
for r being Real
for A being open Subset of REAL
for f being PartFunc of REAL,REAL st f is_differentiable_on A holds
( r (#) f is_differentiable_on A & (r (#) f) `| A = r (#) (f `| A) )
proof
let r be Real; ::_thesis: for A being open Subset of REAL
for f being PartFunc of REAL,REAL st f is_differentiable_on A holds
( r (#) f is_differentiable_on A & (r (#) f) `| A = r (#) (f `| A) )
let A be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st f is_differentiable_on A holds
( r (#) f is_differentiable_on A & (r (#) f) `| A = r (#) (f `| A) )
let f be PartFunc of REAL,REAL; ::_thesis: ( f is_differentiable_on A implies ( r (#) f is_differentiable_on A & (r (#) f) `| A = r (#) (f `| A) ) )
assume A1: f is_differentiable_on A ; ::_thesis: ( r (#) f is_differentiable_on A & (r (#) f) `| A = r (#) (f `| A) )
then A c= dom f by FDIFF_1:def_6;
then A2: A c= dom (r (#) f) by VALUED_1:def_5;
then r (#) f is_differentiable_on A by A1, FDIFF_1:20;
then A3: dom ((r (#) f) `| A) = A by FDIFF_1:def_7;
dom (f `| A) = A by A1, FDIFF_1:def_7;
then A4: dom (r (#) (f `| A)) = A by VALUED_1:def_5;
now__::_thesis:_for_x0_being_Real_st_x0_in_A_holds_
((r_(#)_f)_`|_A)_._x0_=_(r_(#)_(f_`|_A))_._x0
let x0 be Real; ::_thesis: ( x0 in A implies ((r (#) f) `| A) . x0 = (r (#) (f `| A)) . x0 )
assume A5: x0 in A ; ::_thesis: ((r (#) f) `| A) . x0 = (r (#) (f `| A)) . x0
hence ((r (#) f) `| A) . x0 = r * (diff (f,x0)) by A1, A2, FDIFF_1:20
.= r * ((f `| A) . x0) by A1, A5, FDIFF_1:def_7
.= (r (#) (f `| A)) . x0 by A4, A5, VALUED_1:def_5 ;
::_thesis: verum
end;
hence ( r (#) f is_differentiable_on A & (r (#) f) `| A = r (#) (f `| A) ) by A1, A2, A3, A4, FDIFF_1:20, PARTFUN1:5; ::_thesis: verum
end;
theorem :: FDIFF_2:20
for A being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st f1 is_differentiable_on A & f2 is_differentiable_on A holds
( f1 (#) f2 is_differentiable_on A & (f1 (#) f2) `| A = ((f1 `| A) (#) f2) + (f1 (#) (f2 `| A)) )
proof
let A be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_differentiable_on A & f2 is_differentiable_on A holds
( f1 (#) f2 is_differentiable_on A & (f1 (#) f2) `| A = ((f1 `| A) (#) f2) + (f1 (#) (f2 `| A)) )
let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_differentiable_on A & f2 is_differentiable_on A implies ( f1 (#) f2 is_differentiable_on A & (f1 (#) f2) `| A = ((f1 `| A) (#) f2) + (f1 (#) (f2 `| A)) ) )
assume that
A1: f1 is_differentiable_on A and
A2: f2 is_differentiable_on A ; ::_thesis: ( f1 (#) f2 is_differentiable_on A & (f1 (#) f2) `| A = ((f1 `| A) (#) f2) + (f1 (#) (f2 `| A)) )
A3: A c= dom f1 by A1, FDIFF_1:def_6;
A4: A c= dom f2 by A2, FDIFF_1:def_6;
then A c= (dom f1) /\ (dom f2) by A3, XBOOLE_1:19;
then A5: A c= dom (f1 (#) f2) by VALUED_1:def_4;
then f1 (#) f2 is_differentiable_on A by A1, A2, FDIFF_1:21;
then A6: dom ((f1 (#) f2) `| A) = A by FDIFF_1:def_7;
dom (f2 `| A) = A by A2, FDIFF_1:def_7;
then (dom f1) /\ (dom (f2 `| A)) = A by A3, XBOOLE_1:28;
then A7: dom (f1 (#) (f2 `| A)) = A by VALUED_1:def_4;
dom (f1 `| A) = A by A1, FDIFF_1:def_7;
then (dom (f1 `| A)) /\ (dom f2) = A by A4, XBOOLE_1:28;
then dom ((f1 `| A) (#) f2) = A by VALUED_1:def_4;
then (dom ((f1 `| A) (#) f2)) /\ (dom (f1 (#) (f2 `| A))) = A by A7;
then A8: dom (((f1 `| A) (#) f2) + (f1 (#) (f2 `| A))) = A by VALUED_1:def_1;
now__::_thesis:_for_x0_being_Real_st_x0_in_A_holds_
((f1_(#)_f2)_`|_A)_._x0_=_(((f1_`|_A)_(#)_f2)_+_(f1_(#)_(f2_`|_A)))_._x0
let x0 be Real; ::_thesis: ( x0 in A implies ((f1 (#) f2) `| A) . x0 = (((f1 `| A) (#) f2) + (f1 (#) (f2 `| A))) . x0 )
assume A9: x0 in A ; ::_thesis: ((f1 (#) f2) `| A) . x0 = (((f1 `| A) (#) f2) + (f1 (#) (f2 `| A))) . x0
hence ((f1 (#) f2) `| A) . x0 = ((diff (f1,x0)) * (f2 . x0)) + ((f1 . x0) * (diff (f2,x0))) by A1, A2, A5, FDIFF_1:21
.= (((f1 `| A) . x0) * (f2 . x0)) + ((f1 . x0) * (diff (f2,x0))) by A1, A9, FDIFF_1:def_7
.= (((f1 `| A) . x0) * (f2 . x0)) + ((f1 . x0) * ((f2 `| A) . x0)) by A2, A9, FDIFF_1:def_7
.= (((f1 `| A) (#) f2) . x0) + ((f1 . x0) * ((f2 `| A) . x0)) by VALUED_1:5
.= (((f1 `| A) (#) f2) . x0) + ((f1 (#) (f2 `| A)) . x0) by VALUED_1:5
.= (((f1 `| A) (#) f2) + (f1 (#) (f2 `| A))) . x0 by A8, A9, VALUED_1:def_1 ;
::_thesis: verum
end;
hence ( f1 (#) f2 is_differentiable_on A & (f1 (#) f2) `| A = ((f1 `| A) (#) f2) + (f1 (#) (f2 `| A)) ) by A1, A2, A5, A6, A8, FDIFF_1:21, PARTFUN1:5; ::_thesis: verum
end;
Lm3: for f being PartFunc of REAL,REAL holds (f (#) f) " {0} = f " {0}
proof
let f be PartFunc of REAL,REAL; ::_thesis: (f (#) f) " {0} = f " {0}
thus (f (#) f) " {0} c= f " {0} :: according to XBOOLE_0:def_10 ::_thesis: f " {0} c= (f (#) f) " {0}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (f (#) f) " {0} or x in f " {0} )
assume A1: x in (f (#) f) " {0} ; ::_thesis: x in f " {0}
then reconsider x9 = x as Real ;
(f (#) f) . x9 in {0} by A1, FUNCT_1:def_7;
then 0 = (f (#) f) . x9 by TARSKI:def_1
.= (f . x9) * (f . x9) by VALUED_1:5 ;
then f . x9 = 0 ;
then A2: f . x9 in {0} by TARSKI:def_1;
x9 in dom (f (#) f) by A1, FUNCT_1:def_7;
then x9 in (dom f) /\ (dom f) by VALUED_1:def_4;
hence x in f " {0} by A2, FUNCT_1:def_7; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in f " {0} or x in (f (#) f) " {0} )
assume A3: x in f " {0} ; ::_thesis: x in (f (#) f) " {0}
then reconsider x9 = x as Real ;
f . x9 in {0} by A3, FUNCT_1:def_7;
then 0 = f . x9 by TARSKI:def_1;
then (f . x9) * (f . x9) = 0 ;
then (f (#) f) . x9 = 0 by VALUED_1:5;
then A4: (f (#) f) . x9 in {0} by TARSKI:def_1;
x9 in (dom f) /\ (dom f) by A3, FUNCT_1:def_7;
then x9 in dom (f (#) f) by VALUED_1:def_4;
hence x in (f (#) f) " {0} by A4, FUNCT_1:def_7; ::_thesis: verum
end;
theorem :: FDIFF_2:21
for A being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st f1 is_differentiable_on A & f2 is_differentiable_on A & ( for x0 being Real st x0 in A holds
f2 . x0 <> 0 ) holds
( f1 / f2 is_differentiable_on A & (f1 / f2) `| A = (((f1 `| A) (#) f2) - ((f2 `| A) (#) f1)) / (f2 (#) f2) )
proof
let A be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_differentiable_on A & f2 is_differentiable_on A & ( for x0 being Real st x0 in A holds
f2 . x0 <> 0 ) holds
( f1 / f2 is_differentiable_on A & (f1 / f2) `| A = (((f1 `| A) (#) f2) - ((f2 `| A) (#) f1)) / (f2 (#) f2) )
let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_differentiable_on A & f2 is_differentiable_on A & ( for x0 being Real st x0 in A holds
f2 . x0 <> 0 ) implies ( f1 / f2 is_differentiable_on A & (f1 / f2) `| A = (((f1 `| A) (#) f2) - ((f2 `| A) (#) f1)) / (f2 (#) f2) ) )
assume that
A1: f1 is_differentiable_on A and
A2: f2 is_differentiable_on A and
A3: for x0 being Real st x0 in A holds
f2 . x0 <> 0 ; ::_thesis: ( f1 / f2 is_differentiable_on A & (f1 / f2) `| A = (((f1 `| A) (#) f2) - ((f2 `| A) (#) f1)) / (f2 (#) f2) )
A4: A c= dom f1 by A1, FDIFF_1:9;
A5: A c= dom f2 by A2, FDIFF_1:9;
A6: A c= (dom f2) \ (f2 " {0})
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in (dom f2) \ (f2 " {0}) )
assume A7: x in A ; ::_thesis: x in (dom f2) \ (f2 " {0})
then reconsider x9 = x as Real ;
assume not x in (dom f2) \ (f2 " {0}) ; ::_thesis: contradiction
then ( not x in dom f2 or x in f2 " {0} ) by XBOOLE_0:def_5;
then f2 . x9 in {0} by A5, A7, FUNCT_1:def_7;
then f2 . x9 = 0 by TARSKI:def_1;
hence contradiction by A3, A7; ::_thesis: verum
end;
then A c= (dom f1) /\ ((dom f2) \ (f2 " {0})) by A4, XBOOLE_1:19;
then A8: A c= dom (f1 / f2) by RFUNCT_1:def_1;
A9: now__::_thesis:_for_x0_being_Real_st_x0_in_A_holds_
(_f2_._x0_<>_0_&_f1_is_differentiable_in_x0_&_f2_is_differentiable_in_x0_&_f1_/_f2_is_differentiable_in_x0_)
let x0 be Real; ::_thesis: ( x0 in A implies ( f2 . x0 <> 0 & f1 is_differentiable_in x0 & f2 is_differentiable_in x0 & f1 / f2 is_differentiable_in x0 ) )
assume A10: x0 in A ; ::_thesis: ( f2 . x0 <> 0 & f1 is_differentiable_in x0 & f2 is_differentiable_in x0 & f1 / f2 is_differentiable_in x0 )
hence A11: f2 . x0 <> 0 by A3; ::_thesis: ( f1 is_differentiable_in x0 & f2 is_differentiable_in x0 & f1 / f2 is_differentiable_in x0 )
thus A12: f1 is_differentiable_in x0 by A1, A10, FDIFF_1:9; ::_thesis: ( f2 is_differentiable_in x0 & f1 / f2 is_differentiable_in x0 )
thus f2 is_differentiable_in x0 by A2, A10, FDIFF_1:9; ::_thesis: f1 / f2 is_differentiable_in x0
hence f1 / f2 is_differentiable_in x0 by A11, A12, Th14; ::_thesis: verum
end;
then for x0 being Real st x0 in A holds
f1 / f2 is_differentiable_in x0 ;
hence A13: f1 / f2 is_differentiable_on A by A8, FDIFF_1:9; ::_thesis: (f1 / f2) `| A = (((f1 `| A) (#) f2) - ((f2 `| A) (#) f1)) / (f2 (#) f2)
then A14: A = dom ((f1 / f2) `| A) by FDIFF_1:def_7;
A15: now__::_thesis:_for_x0_being_Real_st_x0_in_dom_((f1_/_f2)_`|_A)_holds_
((f1_/_f2)_`|_A)_._x0_=_((((f1_`|_A)_(#)_f2)_-_((f2_`|_A)_(#)_f1))_/_(f2_(#)_f2))_._x0
let x0 be Real; ::_thesis: ( x0 in dom ((f1 / f2) `| A) implies ((f1 / f2) `| A) . x0 = ((((f1 `| A) (#) f2) - ((f2 `| A) (#) f1)) / (f2 (#) f2)) . x0 )
assume A16: x0 in dom ((f1 / f2) `| A) ; ::_thesis: ((f1 / f2) `| A) . x0 = ((((f1 `| A) (#) f2) - ((f2 `| A) (#) f1)) / (f2 (#) f2)) . x0
then A17: f1 is_differentiable_in x0 by A9, A14;
dom (f2 `| A) = A by A2, FDIFF_1:def_7;
then x0 in (dom (f2 `| A)) /\ (dom f1) by A4, A14, A16, XBOOLE_0:def_4;
then A18: x0 in dom ((f2 `| A) (#) f1) by VALUED_1:def_4;
dom (f1 `| A) = A by A1, FDIFF_1:def_7;
then x0 in (dom (f1 `| A)) /\ (dom f2) by A5, A14, A16, XBOOLE_0:def_4;
then x0 in dom ((f1 `| A) (#) f2) by VALUED_1:def_4;
then x0 in (dom ((f1 `| A) (#) f2)) /\ (dom ((f2 `| A) (#) f1)) by A18, XBOOLE_0:def_4;
then A19: x0 in dom (((f1 `| A) (#) f2) - ((f2 `| A) (#) f1)) by VALUED_1:12;
A20: f2 . x0 <> 0 by A9, A14, A16;
then (f2 . x0) * (f2 . x0) <> 0 ;
then (f2 (#) f2) . x0 <> 0 by VALUED_1:5;
then not (f2 (#) f2) . x0 in {0} by TARSKI:def_1;
then A21: not x0 in (f2 (#) f2) " {0} by FUNCT_1:def_7;
x0 in (dom f2) /\ (dom f2) by A5, A14, A16;
then x0 in dom (f2 (#) f2) by VALUED_1:def_4;
then x0 in (dom (f2 (#) f2)) \ ((f2 (#) f2) " {0}) by A21, XBOOLE_0:def_5;
then x0 in (dom (((f1 `| A) (#) f2) - ((f2 `| A) (#) f1))) /\ ((dom (f2 (#) f2)) \ ((f2 (#) f2) " {0})) by A19, XBOOLE_0:def_4;
then A22: x0 in dom ((((f1 `| A) (#) f2) - ((f2 `| A) (#) f1)) / (f2 (#) f2)) by RFUNCT_1:def_1;
A23: f2 is_differentiable_in x0 by A9, A14, A16;
thus ((f1 / f2) `| A) . x0 = diff ((f1 / f2),x0) by A13, A14, A16, FDIFF_1:def_7
.= (((diff (f1,x0)) * (f2 . x0)) - ((diff (f2,x0)) * (f1 . x0))) / ((f2 . x0) ^2) by A20, A17, A23, Th14
.= ((((f1 `| A) . x0) * (f2 . x0)) - ((diff (f2,x0)) * (f1 . x0))) / ((f2 . x0) ^2) by A1, A14, A16, FDIFF_1:def_7
.= ((((f1 `| A) . x0) * (f2 . x0)) - (((f2 `| A) . x0) * (f1 . x0))) / ((f2 . x0) ^2) by A2, A14, A16, FDIFF_1:def_7
.= ((((f1 `| A) (#) f2) . x0) - (((f2 `| A) . x0) * (f1 . x0))) / ((f2 . x0) ^2) by VALUED_1:5
.= ((((f1 `| A) (#) f2) . x0) - (((f2 `| A) (#) f1) . x0)) / ((f2 . x0) ^2) by VALUED_1:5
.= ((((f1 `| A) (#) f2) - ((f2 `| A) (#) f1)) . x0) / ((f2 . x0) * (f2 . x0)) by A19, VALUED_1:13
.= ((((f1 `| A) (#) f2) - ((f2 `| A) (#) f1)) . x0) / ((f2 (#) f2) . x0) by VALUED_1:5
.= ((((f1 `| A) (#) f2) - ((f2 `| A) (#) f1)) . x0) * (((f2 (#) f2) . x0) ") by XCMPLX_0:def_9
.= ((((f1 `| A) (#) f2) - ((f2 `| A) (#) f1)) / (f2 (#) f2)) . x0 by A22, RFUNCT_1:def_1 ; ::_thesis: verum
end;
dom ((((f1 `| A) (#) f2) - ((f2 `| A) (#) f1)) / (f2 (#) f2)) = (dom (((f1 `| A) (#) f2) - ((f2 `| A) (#) f1))) /\ ((dom (f2 (#) f2)) \ ((f2 (#) f2) " {0})) by RFUNCT_1:def_1
.= ((dom ((f1 `| A) (#) f2)) /\ (dom ((f2 `| A) (#) f1))) /\ ((dom (f2 (#) f2)) \ ((f2 (#) f2) " {0})) by VALUED_1:12
.= (((dom (f1 `| A)) /\ (dom f2)) /\ (dom ((f2 `| A) (#) f1))) /\ ((dom (f2 (#) f2)) \ ((f2 (#) f2) " {0})) by VALUED_1:def_4
.= ((A /\ (dom f2)) /\ (dom ((f2 `| A) (#) f1))) /\ ((dom (f2 (#) f2)) \ ((f2 (#) f2) " {0})) by A1, FDIFF_1:def_7
.= (A /\ (dom ((f2 `| A) (#) f1))) /\ ((dom (f2 (#) f2)) \ ((f2 (#) f2) " {0})) by A5, XBOOLE_1:28
.= (A /\ ((dom (f2 `| A)) /\ (dom f1))) /\ ((dom (f2 (#) f2)) \ ((f2 (#) f2) " {0})) by VALUED_1:def_4
.= (A /\ (A /\ (dom f1))) /\ ((dom (f2 (#) f2)) \ ((f2 (#) f2) " {0})) by A2, FDIFF_1:def_7
.= (A /\ A) /\ ((dom (f2 (#) f2)) \ ((f2 (#) f2) " {0})) by A4, XBOOLE_1:28
.= A /\ (((dom f2) /\ (dom f2)) \ ((f2 (#) f2) " {0})) by VALUED_1:def_4
.= A /\ ((dom f2) \ (f2 " {0})) by Lm3
.= dom ((f1 / f2) `| A) by A6, A14, XBOOLE_1:28 ;
hence (f1 / f2) `| A = (((f1 `| A) (#) f2) - ((f2 `| A) (#) f1)) / (f2 (#) f2) by A15, PARTFUN1:5; ::_thesis: verum
end;
theorem :: FDIFF_2:22
for A being open Subset of REAL
for f being PartFunc of REAL,REAL st f is_differentiable_on A & ( for x0 being Real st x0 in A holds
f . x0 <> 0 ) holds
( f ^ is_differentiable_on A & (f ^) `| A = - ((f `| A) / (f (#) f)) )
proof
let A be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st f is_differentiable_on A & ( for x0 being Real st x0 in A holds
f . x0 <> 0 ) holds
( f ^ is_differentiable_on A & (f ^) `| A = - ((f `| A) / (f (#) f)) )
let f be PartFunc of REAL,REAL; ::_thesis: ( f is_differentiable_on A & ( for x0 being Real st x0 in A holds
f . x0 <> 0 ) implies ( f ^ is_differentiable_on A & (f ^) `| A = - ((f `| A) / (f (#) f)) ) )
assume that
A1: f is_differentiable_on A and
A2: for x0 being Real st x0 in A holds
f . x0 <> 0 ; ::_thesis: ( f ^ is_differentiable_on A & (f ^) `| A = - ((f `| A) / (f (#) f)) )
A3: now__::_thesis:_for_x0_being_Real_st_x0_in_A_holds_
(_f_._x0_<>_0_&_f_is_differentiable_in_x0_&_f_^_is_differentiable_in_x0_)
let x0 be Real; ::_thesis: ( x0 in A implies ( f . x0 <> 0 & f is_differentiable_in x0 & f ^ is_differentiable_in x0 ) )
assume A4: x0 in A ; ::_thesis: ( f . x0 <> 0 & f is_differentiable_in x0 & f ^ is_differentiable_in x0 )
hence A5: f . x0 <> 0 by A2; ::_thesis: ( f is_differentiable_in x0 & f ^ is_differentiable_in x0 )
thus f is_differentiable_in x0 by A1, A4, FDIFF_1:9; ::_thesis: f ^ is_differentiable_in x0
hence f ^ is_differentiable_in x0 by A5, Th15; ::_thesis: verum
end;
then A6: for x0 being Real st x0 in A holds
f ^ is_differentiable_in x0 ;
A7: A c= dom f by A1, FDIFF_1:9;
A8: A c= (dom f) \ (f " {0})
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in (dom f) \ (f " {0}) )
assume A9: x in A ; ::_thesis: x in (dom f) \ (f " {0})
then reconsider x9 = x as Real ;
assume not x in (dom f) \ (f " {0}) ; ::_thesis: contradiction
then ( not x in dom f or x in f " {0} ) by XBOOLE_0:def_5;
then f . x9 in {0} by A7, A9, FUNCT_1:def_7;
then f . x9 = 0 by TARSKI:def_1;
hence contradiction by A2, A9; ::_thesis: verum
end;
then A c= dom (f ^) by RFUNCT_1:def_2;
hence A10: f ^ is_differentiable_on A by A6, FDIFF_1:9; ::_thesis: (f ^) `| A = - ((f `| A) / (f (#) f))
then A11: A = dom ((f ^) `| A) by FDIFF_1:def_7;
A12: now__::_thesis:_for_x0_being_Real_st_x0_in_dom_((f_^)_`|_A)_holds_
((f_^)_`|_A)_._x0_=_(-_((f_`|_A)_/_(f_(#)_f)))_._x0
let x0 be Real; ::_thesis: ( x0 in dom ((f ^) `| A) implies ((f ^) `| A) . x0 = (- ((f `| A) / (f (#) f))) . x0 )
A13: dom (f `| A) = A by A1, FDIFF_1:def_7;
assume A14: x0 in dom ((f ^) `| A) ; ::_thesis: ((f ^) `| A) . x0 = (- ((f `| A) / (f (#) f))) . x0
then A15: f is_differentiable_in x0 by A3, A11;
A16: f . x0 <> 0 by A3, A11, A14;
then (f . x0) * (f . x0) <> 0 ;
then (f (#) f) . x0 <> 0 by VALUED_1:5;
then not (f (#) f) . x0 in {0} by TARSKI:def_1;
then A17: not x0 in (f (#) f) " {0} by FUNCT_1:def_7;
x0 in (dom f) /\ (dom f) by A7, A11, A14;
then x0 in dom (f (#) f) by VALUED_1:def_4;
then x0 in (dom (f (#) f)) \ ((f (#) f) " {0}) by A17, XBOOLE_0:def_5;
then x0 in (dom (f `| A)) /\ ((dom (f (#) f)) \ ((f (#) f) " {0})) by A11, A14, A13, XBOOLE_0:def_4;
then A18: x0 in dom ((f `| A) / (f (#) f)) by RFUNCT_1:def_1;
thus ((f ^) `| A) . x0 = diff ((f ^),x0) by A10, A11, A14, FDIFF_1:def_7
.= - ((diff (f,x0)) / ((f . x0) ^2)) by A16, A15, Th15
.= - (((f `| A) . x0) / ((f . x0) * (f . x0))) by A1, A11, A14, FDIFF_1:def_7
.= - (((f `| A) . x0) / ((f (#) f) . x0)) by VALUED_1:5
.= - (((f `| A) . x0) * (((f (#) f) . x0) ")) by XCMPLX_0:def_9
.= - (((f `| A) / (f (#) f)) . x0) by A18, RFUNCT_1:def_1
.= (- ((f `| A) / (f (#) f))) . x0 by VALUED_1:8 ; ::_thesis: verum
end;
dom (- ((f `| A) / (f (#) f))) = dom ((f `| A) / (f (#) f)) by VALUED_1:8
.= (dom (f `| A)) /\ ((dom (f (#) f)) \ ((f (#) f) " {0})) by RFUNCT_1:def_1
.= A /\ ((dom (f (#) f)) \ ((f (#) f) " {0})) by A1, FDIFF_1:def_7
.= A /\ (((dom f) /\ (dom f)) \ ((f (#) f) " {0})) by VALUED_1:def_4
.= A /\ ((dom f) \ (f " {0})) by Lm3
.= dom ((f ^) `| A) by A8, A11, XBOOLE_1:28 ;
hence (f ^) `| A = - ((f `| A) / (f (#) f)) by A12, PARTFUN1:5; ::_thesis: verum
end;
theorem :: FDIFF_2:23
for A being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st f1 is_differentiable_on A & f1 .: A is open Subset of REAL & f2 is_differentiable_on f1 .: A holds
( f2 * f1 is_differentiable_on A & (f2 * f1) `| A = ((f2 `| (f1 .: A)) * f1) (#) (f1 `| A) )
proof
let A be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_differentiable_on A & f1 .: A is open Subset of REAL & f2 is_differentiable_on f1 .: A holds
( f2 * f1 is_differentiable_on A & (f2 * f1) `| A = ((f2 `| (f1 .: A)) * f1) (#) (f1 `| A) )
let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is_differentiable_on A & f1 .: A is open Subset of REAL & f2 is_differentiable_on f1 .: A implies ( f2 * f1 is_differentiable_on A & (f2 * f1) `| A = ((f2 `| (f1 .: A)) * f1) (#) (f1 `| A) ) )
assume that
A1: f1 is_differentiable_on A and
A2: f1 .: A is open Subset of REAL and
A3: f2 is_differentiable_on f1 .: A ; ::_thesis: ( f2 * f1 is_differentiable_on A & (f2 * f1) `| A = ((f2 `| (f1 .: A)) * f1) (#) (f1 `| A) )
A4: A c= dom f1 by A1, FDIFF_1:9;
A5: now__::_thesis:_for_x0_being_Real_st_x0_in_A_holds_
(_f1_is_differentiable_in_x0_&_x0_in_dom_f1_&_f1_._x0_in_f1_.:_A_&_f2_is_differentiable_in_f1_._x0_&_f2_*_f1_is_differentiable_in_x0_)
let x0 be Real; ::_thesis: ( x0 in A implies ( f1 is_differentiable_in x0 & x0 in dom f1 & f1 . x0 in f1 .: A & f2 is_differentiable_in f1 . x0 & f2 * f1 is_differentiable_in x0 ) )
assume A6: x0 in A ; ::_thesis: ( f1 is_differentiable_in x0 & x0 in dom f1 & f1 . x0 in f1 .: A & f2 is_differentiable_in f1 . x0 & f2 * f1 is_differentiable_in x0 )
hence A7: f1 is_differentiable_in x0 by A1, FDIFF_1:9; ::_thesis: ( x0 in dom f1 & f1 . x0 in f1 .: A & f2 is_differentiable_in f1 . x0 & f2 * f1 is_differentiable_in x0 )
thus x0 in dom f1 by A4, A6; ::_thesis: ( f1 . x0 in f1 .: A & f2 is_differentiable_in f1 . x0 & f2 * f1 is_differentiable_in x0 )
thus f1 . x0 in f1 .: A by A4, A6, FUNCT_1:def_6; ::_thesis: ( f2 is_differentiable_in f1 . x0 & f2 * f1 is_differentiable_in x0 )
hence f2 is_differentiable_in f1 . x0 by A2, A3, FDIFF_1:9; ::_thesis: f2 * f1 is_differentiable_in x0
hence f2 * f1 is_differentiable_in x0 by A7, Th13; ::_thesis: verum
end;
f1 .: A c= dom f2 by A2, A3, FDIFF_1:9;
then A8: A c= dom (f2 * f1) by A4, FUNCT_3:3;
for x0 being Real st x0 in A holds
f2 * f1 is_differentiable_in x0 by A5;
hence A9: f2 * f1 is_differentiable_on A by A8, FDIFF_1:9; ::_thesis: (f2 * f1) `| A = ((f2 `| (f1 .: A)) * f1) (#) (f1 `| A)
then A10: dom ((f2 * f1) `| A) = A by FDIFF_1:def_7;
A11: now__::_thesis:_for_x0_being_Real_st_x0_in_dom_((f2_*_f1)_`|_A)_holds_
((f2_*_f1)_`|_A)_._x0_=_(((f2_`|_(f1_.:_A))_*_f1)_(#)_(f1_`|_A))_._x0
let x0 be Real; ::_thesis: ( x0 in dom ((f2 * f1) `| A) implies ((f2 * f1) `| A) . x0 = (((f2 `| (f1 .: A)) * f1) (#) (f1 `| A)) . x0 )
assume A12: x0 in dom ((f2 * f1) `| A) ; ::_thesis: ((f2 * f1) `| A) . x0 = (((f2 `| (f1 .: A)) * f1) (#) (f1 `| A)) . x0
then A13: f1 is_differentiable_in x0 by A5, A10;
A14: x0 in dom f1 by A5, A10, A12;
A15: f1 . x0 in f1 .: A by A5, A10, A12;
A16: f2 is_differentiable_in f1 . x0 by A5, A10, A12;
thus ((f2 * f1) `| A) . x0 = diff ((f2 * f1),x0) by A9, A10, A12, FDIFF_1:def_7
.= (diff (f2,(f1 . x0))) * (diff (f1,x0)) by A13, A16, Th13
.= (diff (f2,(f1 . x0))) * ((f1 `| A) . x0) by A1, A10, A12, FDIFF_1:def_7
.= ((f2 `| (f1 .: A)) . (f1 . x0)) * ((f1 `| A) . x0) by A3, A15, FDIFF_1:def_7
.= (((f2 `| (f1 .: A)) * f1) . x0) * ((f1 `| A) . x0) by A14, FUNCT_1:13
.= (((f2 `| (f1 .: A)) * f1) (#) (f1 `| A)) . x0 by VALUED_1:5 ; ::_thesis: verum
end;
dom (f2 `| (f1 .: A)) = f1 .: A by A3, FDIFF_1:def_7;
then A c= dom ((f2 `| (f1 .: A)) * f1) by A4, FUNCT_1:101;
then dom ((f2 * f1) `| A) = (dom ((f2 `| (f1 .: A)) * f1)) /\ A by A10, XBOOLE_1:28
.= (dom ((f2 `| (f1 .: A)) * f1)) /\ (dom (f1 `| A)) by A1, FDIFF_1:def_7
.= dom (((f2 `| (f1 .: A)) * f1) (#) (f1 `| A)) by VALUED_1:def_4 ;
hence (f2 * f1) `| A = ((f2 `| (f1 .: A)) * f1) (#) (f1 `| A) by A11, PARTFUN1:5; ::_thesis: verum
end;
theorem Th24: :: FDIFF_2:24
for A being open Subset of REAL
for f being PartFunc of REAL,REAL st A c= dom f & ( for r, p being Real st r in A & p in A holds
abs ((f . r) - (f . p)) <= (r - p) ^2 ) holds
( f is_differentiable_on A & ( for x0 being Real st x0 in A holds
diff (f,x0) = 0 ) )
proof
let A be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st A c= dom f & ( for r, p being Real st r in A & p in A holds
abs ((f . r) - (f . p)) <= (r - p) ^2 ) holds
( f is_differentiable_on A & ( for x0 being Real st x0 in A holds
diff (f,x0) = 0 ) )
let f be PartFunc of REAL,REAL; ::_thesis: ( A c= dom f & ( for r, p being Real st r in A & p in A holds
abs ((f . r) - (f . p)) <= (r - p) ^2 ) implies ( f is_differentiable_on A & ( for x0 being Real st x0 in A holds
diff (f,x0) = 0 ) ) )
assume that
A1: A c= dom f and
A2: for r, p being Real st r in A & p in A holds
abs ((f . r) - (f . p)) <= (r - p) ^2 ; ::_thesis: ( f is_differentiable_on A & ( for x0 being Real st x0 in A holds
diff (f,x0) = 0 ) )
A3: now__::_thesis:_for_x0_being_Real_st_x0_in_A_holds_
(_f_is_differentiable_in_x0_&_diff_(f,x0)_=_0_)
let x0 be Real; ::_thesis: ( x0 in A implies ( f is_differentiable_in x0 & diff (f,x0) = 0 ) )
assume x0 in A ; ::_thesis: ( f is_differentiable_in x0 & diff (f,x0) = 0 )
then consider N being Neighbourhood of x0 such that
A4: N c= A by RCOMP_1:18;
A5: N c= dom f by A1, A4, XBOOLE_1:1;
for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = 0 )
proof
reconsider a = NAT --> 0 as Real_Sequence by FUNCOP_1:45;
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = 0 )
let c be V8() Real_Sequence; ::_thesis: ( rng c = {x0} & rng (h + c) c= dom f implies ( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = 0 ) )
assume that
A6: rng c = {x0} and
A7: rng (h + c) c= dom f ; ::_thesis: ( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = 0 )
A8: lim (h + c) = x0 by A6, Th4;
consider r being real number such that
A9: 0 < r and
A10: N = ].(x0 - r),(x0 + r).[ by RCOMP_1:def_6;
h + c is convergent by A6, Th4;
then consider n being Element of NAT such that
A11: for m being Element of NAT st n <= m holds
abs (((h + c) . m) - x0) < r by A8, A9, SEQ_2:def_7;
set hc = (h + c) ^\ n;
A12: rng ((h + c) ^\ n) c= A
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng ((h + c) ^\ n) or x in A )
assume x in rng ((h + c) ^\ n) ; ::_thesis: x in A
then consider m being Element of NAT such that
A13: x = ((h + c) ^\ n) . m by FUNCT_2:113;
x = (h + c) . (m + n) by A13, NAT_1:def_3;
then abs ((((h + c) ^\ n) . m) - x0) < r by A11, A13, NAT_1:12;
then x in N by A10, A13, RCOMP_1:1;
hence x in A by A4; ::_thesis: verum
end;
A14: rng c c= dom f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng c or x in dom f )
assume x in rng c ; ::_thesis: x in dom f
then A15: x = x0 by A6, TARSKI:def_1;
x0 in N by RCOMP_1:16;
hence x in dom f by A5, A15; ::_thesis: verum
end;
set fn = ((h ") (#) ((f /* (h + c)) - (f /* c))) ^\ n;
set h1 = h ^\ n;
set c1 = c ^\ n;
A18: rng (c ^\ n) c= A
proof
A19: rng (c ^\ n) c= rng c by VALUED_0:21;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (c ^\ n) or x in A )
assume x in rng (c ^\ n) ; ::_thesis: x in A
then A20: x = x0 by A6, A19, TARSKI:def_1;
x0 in N by RCOMP_1:16;
hence x in A by A4, A20; ::_thesis: verum
end;
A21: abs (h ^\ n) is non-zero by SEQ_1:53;
A22: for m being Element of NAT holds
( a . m <= (abs (((h ") (#) ((f /* (h + c)) - (f /* c))) ^\ n)) . m & (abs (((h ") (#) ((f /* (h + c)) - (f /* c))) ^\ n)) . m <= (abs (h ^\ n)) . m )
proof
let m be Element of NAT ; ::_thesis: ( a . m <= (abs (((h ") (#) ((f /* (h + c)) - (f /* c))) ^\ n)) . m & (abs (((h ") (#) ((f /* (h + c)) - (f /* c))) ^\ n)) . m <= (abs (h ^\ n)) . m )
A23: (c ^\ n) . m in rng (c ^\ n) by VALUED_0:28;
((h + c) ^\ n) . m in rng ((h + c) ^\ n) by VALUED_0:28;
then abs ((f . (((h + c) ^\ n) . m)) - (f . ((c ^\ n) . m))) <= ((((h + c) ^\ n) . m) - ((c ^\ n) . m)) ^2 by A2, A12, A18, A23;
then A24: abs ((f . (((h + c) ^\ n) . m)) - (f . ((c ^\ n) . m))) <= (abs ((((h + c) ^\ n) . m) - ((c ^\ n) . m))) ^2 by COMPLEX1:75;
A25: (((abs (h ^\ n)) . m) ") * ((abs (((f /* (h + c)) - (f /* c)) ^\ n)) . m) = (((abs (h ^\ n)) ") . m) * ((abs (((f /* (h + c)) - (f /* c)) ^\ n)) . m) by VALUED_1:10
.= (((abs (h ^\ n)) ") (#) (abs (((f /* (h + c)) - (f /* c)) ^\ n))) . m by SEQ_1:8
.= ((abs ((h ^\ n) ")) (#) (abs (((f /* (h + c)) - (f /* c)) ^\ n))) . m by SEQ_1:54
.= (abs (((h ^\ n) ") (#) (((f /* (h + c)) - (f /* c)) ^\ n))) . m by SEQ_1:52
.= (abs (((h ") ^\ n) (#) (((f /* (h + c)) - (f /* c)) ^\ n))) . m by SEQM_3:18
.= (abs (((h ") (#) ((f /* (h + c)) - (f /* c))) ^\ n)) . m by SEQM_3:19 ;
0 <= abs ((((h ") (#) ((f /* (h + c)) - (f /* c))) ^\ n) . m) by COMPLEX1:46;
then a . m <= abs ((((h ") (#) ((f /* (h + c)) - (f /* c))) ^\ n) . m) by FUNCOP_1:7;
hence a . m <= (abs (((h ") (#) ((f /* (h + c)) - (f /* c))) ^\ n)) . m by SEQ_1:12; ::_thesis: (abs (((h ") (#) ((f /* (h + c)) - (f /* c))) ^\ n)) . m <= (abs (h ^\ n)) . m
A26: (abs ((((h + c) ^\ n) . m) - ((c ^\ n) . m))) ^2 = (abs ((((h ^\ n) + (c ^\ n)) . m) - ((c ^\ n) . m))) ^2 by SEQM_3:15
.= (abs ((((h ^\ n) . m) + ((c ^\ n) . m)) - ((c ^\ n) . m))) ^2 by SEQ_1:7
.= ((abs (h ^\ n)) . m) ^2 by SEQ_1:12
.= ((abs (h ^\ n)) . m) * ((abs (h ^\ n)) . m) ;
0 <= abs ((h ^\ n) . m) by COMPLEX1:46;
then A27: 0 <= (abs (h ^\ n)) . m by SEQ_1:12;
A28: (abs (h ^\ n)) . m <> 0 by A21, SEQ_1:5;
A29: (((abs (h ^\ n)) . m) * ((abs (h ^\ n)) . m)) * (((abs (h ^\ n)) . m) ") = ((abs (h ^\ n)) . m) * (((abs (h ^\ n)) . m) * (((abs (h ^\ n)) . m) "))
.= ((abs (h ^\ n)) . m) * 1 by A28, XCMPLX_0:def_7
.= (abs (h ^\ n)) . m ;
abs ((f . (((h + c) ^\ n) . m)) - (f . ((c ^\ n) . m))) = abs (((f /* ((h + c) ^\ n)) . m) - (f . ((c ^\ n) . m))) by A1, A12, FUNCT_2:108, XBOOLE_1:1
.= abs (((f /* ((h + c) ^\ n)) . m) - ((f /* (c ^\ n)) . m)) by A1, A18, FUNCT_2:108, XBOOLE_1:1
.= abs (((f /* ((h + c) ^\ n)) - (f /* (c ^\ n))) . m) by RFUNCT_2:1
.= (abs ((f /* ((h + c) ^\ n)) - (f /* (c ^\ n)))) . m by SEQ_1:12
.= (abs (((f /* (h + c)) ^\ n) - (f /* (c ^\ n)))) . m by A7, VALUED_0:27
.= (abs (((f /* (h + c)) ^\ n) - ((f /* c) ^\ n))) . m by A14, VALUED_0:27
.= (abs (((f /* (h + c)) - (f /* c)) ^\ n)) . m by SEQM_3:17 ;
hence (abs (((h ") (#) ((f /* (h + c)) - (f /* c))) ^\ n)) . m <= (abs (h ^\ n)) . m by A24, A26, A27, A29, A25, XREAL_1:64; ::_thesis: verum
end;
lim (h ^\ n) = 0 ;
then A30: lim (abs (h ^\ n)) = abs 0 by SEQ_4:14
.= 0 by ABSVALUE:2 ;
A31: lim a = a . 0 by SEQ_4:26
.= 0 by FUNCOP_1:7 ;
then A32: lim (abs (((h ") (#) ((f /* (h + c)) - (f /* c))) ^\ n)) = 0 by A30, A22, SEQ_2:20;
A33: abs (((h ") (#) ((f /* (h + c)) - (f /* c))) ^\ n) is convergent by A31, A30, A22, SEQ_2:19;
then A34: ((h ") (#) ((f /* (h + c)) - (f /* c))) ^\ n is convergent by A32, SEQ_4:15;
hence (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent by SEQ_4:21; ::_thesis: lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = 0
lim (((h ") (#) ((f /* (h + c)) - (f /* c))) ^\ n) = 0 by A33, A32, SEQ_4:15;
hence lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = 0 by A34, SEQ_4:22; ::_thesis: verum
end;
hence ( f is_differentiable_in x0 & diff (f,x0) = 0 ) by A5, Th12; ::_thesis: verum
end;
then for x0 being Real st x0 in A holds
f is_differentiable_in x0 ;
hence f is_differentiable_on A by A1, FDIFF_1:9; ::_thesis: for x0 being Real st x0 in A holds
diff (f,x0) = 0
let x0 be Real; ::_thesis: ( x0 in A implies diff (f,x0) = 0 )
assume x0 in A ; ::_thesis: diff (f,x0) = 0
hence diff (f,x0) = 0 by A3; ::_thesis: verum
end;
theorem Th25: :: FDIFF_2:25
for p, g being Real
for f being PartFunc of REAL,REAL st ( for r1, r2 being Real st r1 in ].p,g.[ & r2 in ].p,g.[ holds
abs ((f . r1) - (f . r2)) <= (r1 - r2) ^2 ) & ].p,g.[ c= dom f holds
( f is_differentiable_on ].p,g.[ & f | ].p,g.[ is V8() )
proof
let p, g be Real; ::_thesis: for f being PartFunc of REAL,REAL st ( for r1, r2 being Real st r1 in ].p,g.[ & r2 in ].p,g.[ holds
abs ((f . r1) - (f . r2)) <= (r1 - r2) ^2 ) & ].p,g.[ c= dom f holds
( f is_differentiable_on ].p,g.[ & f | ].p,g.[ is V8() )
let f be PartFunc of REAL,REAL; ::_thesis: ( ( for r1, r2 being Real st r1 in ].p,g.[ & r2 in ].p,g.[ holds
abs ((f . r1) - (f . r2)) <= (r1 - r2) ^2 ) & ].p,g.[ c= dom f implies ( f is_differentiable_on ].p,g.[ & f | ].p,g.[ is V8() ) )
assume that
A1: for r1, r2 being Real st r1 in ].p,g.[ & r2 in ].p,g.[ holds
abs ((f . r1) - (f . r2)) <= (r1 - r2) ^2 and
A2: ].p,g.[ c= dom f ; ::_thesis: ( f is_differentiable_on ].p,g.[ & f | ].p,g.[ is V8() )
thus A3: f is_differentiable_on ].p,g.[ by A1, A2, Th24; ::_thesis: f | ].p,g.[ is V8()
for x0 being Real st x0 in ].p,g.[ holds
diff (f,x0) = 0 by A1, A2, Th24;
hence f | ].p,g.[ is V8() by A2, A3, ROLLE:7; ::_thesis: verum
end;
theorem :: FDIFF_2:26
for r being Real
for f being PartFunc of REAL,REAL st left_open_halfline r c= dom f & ( for r1, r2 being Real st r1 in left_open_halfline r & r2 in left_open_halfline r holds
abs ((f . r1) - (f . r2)) <= (r1 - r2) ^2 ) holds
( f is_differentiable_on left_open_halfline r & f | (left_open_halfline r) is V8() )
proof
let r be Real; ::_thesis: for f being PartFunc of REAL,REAL st left_open_halfline r c= dom f & ( for r1, r2 being Real st r1 in left_open_halfline r & r2 in left_open_halfline r holds
abs ((f . r1) - (f . r2)) <= (r1 - r2) ^2 ) holds
( f is_differentiable_on left_open_halfline r & f | (left_open_halfline r) is V8() )
let f be PartFunc of REAL,REAL; ::_thesis: ( left_open_halfline r c= dom f & ( for r1, r2 being Real st r1 in left_open_halfline r & r2 in left_open_halfline r holds
abs ((f . r1) - (f . r2)) <= (r1 - r2) ^2 ) implies ( f is_differentiable_on left_open_halfline r & f | (left_open_halfline r) is V8() ) )
assume that
A1: left_open_halfline r c= dom f and
A2: for r1, r2 being Real st r1 in left_open_halfline r & r2 in left_open_halfline r holds
abs ((f . r1) - (f . r2)) <= (r1 - r2) ^2 ; ::_thesis: ( f is_differentiable_on left_open_halfline r & f | (left_open_halfline r) is V8() )
now__::_thesis:_for_r1,_r2_being_Real_st_r1_in_(left_open_halfline_r)_/\_(dom_f)_&_r2_in_(left_open_halfline_r)_/\_(dom_f)_holds_
f_._r1_=_f_._r2
let r1, r2 be Real; ::_thesis: ( r1 in (left_open_halfline r) /\ (dom f) & r2 in (left_open_halfline r) /\ (dom f) implies f . r1 = f . r2 )
assume that
A3: r1 in (left_open_halfline r) /\ (dom f) and
A4: r2 in (left_open_halfline r) /\ (dom f) ; ::_thesis: f . r1 = f . r2
set rr = min (r1,r2);
A5: ].((min (r1,r2)) - 1),r.[ c= left_open_halfline r by XXREAL_1:263;
then A6: for g1, g2 being Real st g1 in ].((min (r1,r2)) - 1),r.[ & g2 in ].((min (r1,r2)) - 1),r.[ holds
abs ((f . g1) - (f . g2)) <= (g1 - g2) ^2 by A2;
r2 in left_open_halfline r by A4, XBOOLE_0:def_4;
then r2 in { p where p is Real : p < r } by XXREAL_1:229;
then A7: ex g2 being Real st
( g2 = r2 & g2 < r ) ;
(min (r1,r2)) - 1 < r2 - 0 by XREAL_1:15, XXREAL_0:17;
then r2 in { g2 where g2 is Real : ( (min (r1,r2)) - 1 < g2 & g2 < r ) } by A7;
then A8: r2 in ].((min (r1,r2)) - 1),r.[ by RCOMP_1:def_2;
r2 in dom f by A4, XBOOLE_0:def_4;
then A9: r2 in ].((min (r1,r2)) - 1),r.[ /\ (dom f) by A8, XBOOLE_0:def_4;
r1 in left_open_halfline r by A3, XBOOLE_0:def_4;
then r1 in { g where g is Real : g < r } by XXREAL_1:229;
then A10: ex g1 being Real st
( g1 = r1 & g1 < r ) ;
(min (r1,r2)) - 1 < r1 - 0 by XREAL_1:15, XXREAL_0:17;
then r1 in { g1 where g1 is Real : ( (min (r1,r2)) - 1 < g1 & g1 < r ) } by A10;
then A11: r1 in ].((min (r1,r2)) - 1),r.[ by RCOMP_1:def_2;
r1 in dom f by A3, XBOOLE_0:def_4;
then A12: r1 in ].((min (r1,r2)) - 1),r.[ /\ (dom f) by A11, XBOOLE_0:def_4;
].((min (r1,r2)) - 1),r.[ c= dom f by A1, A5, XBOOLE_1:1;
then f | ].((min (r1,r2)) - 1),r.[ is V8() by A6, Th25;
hence f . r1 = f . r2 by A12, A9, PARTFUN2:58; ::_thesis: verum
end;
hence ( f is_differentiable_on left_open_halfline r & f | (left_open_halfline r) is V8() ) by A1, A2, Th24, PARTFUN2:58; ::_thesis: verum
end;
theorem :: FDIFF_2:27
for r being Real
for f being PartFunc of REAL,REAL st right_open_halfline r c= dom f & ( for r1, r2 being Real st r1 in right_open_halfline r & r2 in right_open_halfline r holds
abs ((f . r1) - (f . r2)) <= (r1 - r2) ^2 ) holds
( f is_differentiable_on right_open_halfline r & f | (right_open_halfline r) is V8() )
proof
let r be Real; ::_thesis: for f being PartFunc of REAL,REAL st right_open_halfline r c= dom f & ( for r1, r2 being Real st r1 in right_open_halfline r & r2 in right_open_halfline r holds
abs ((f . r1) - (f . r2)) <= (r1 - r2) ^2 ) holds
( f is_differentiable_on right_open_halfline r & f | (right_open_halfline r) is V8() )
let f be PartFunc of REAL,REAL; ::_thesis: ( right_open_halfline r c= dom f & ( for r1, r2 being Real st r1 in right_open_halfline r & r2 in right_open_halfline r holds
abs ((f . r1) - (f . r2)) <= (r1 - r2) ^2 ) implies ( f is_differentiable_on right_open_halfline r & f | (right_open_halfline r) is V8() ) )
assume that
A1: right_open_halfline r c= dom f and
A2: for r1, r2 being Real st r1 in right_open_halfline r & r2 in right_open_halfline r holds
abs ((f . r1) - (f . r2)) <= (r1 - r2) ^2 ; ::_thesis: ( f is_differentiable_on right_open_halfline r & f | (right_open_halfline r) is V8() )
now__::_thesis:_for_r1,_r2_being_Real_st_r1_in_(right_open_halfline_r)_/\_(dom_f)_&_r2_in_(right_open_halfline_r)_/\_(dom_f)_holds_
f_._r1_=_f_._r2
let r1, r2 be Real; ::_thesis: ( r1 in (right_open_halfline r) /\ (dom f) & r2 in (right_open_halfline r) /\ (dom f) implies f . r1 = f . r2 )
assume that
A3: r1 in (right_open_halfline r) /\ (dom f) and
A4: r2 in (right_open_halfline r) /\ (dom f) ; ::_thesis: f . r1 = f . r2
set rr = max (r1,r2);
A5: ].r,((max (r1,r2)) + 1).[ c= right_open_halfline r by XXREAL_1:247;
then A6: for g1, g2 being Real st g1 in ].r,((max (r1,r2)) + 1).[ & g2 in ].r,((max (r1,r2)) + 1).[ holds
abs ((f . g1) - (f . g2)) <= (g1 - g2) ^2 by A2;
r2 in right_open_halfline r by A4, XBOOLE_0:def_4;
then r2 in { p where p is Real : r < p } by XXREAL_1:230;
then A7: ex g2 being Real st
( g2 = r2 & r < g2 ) ;
r2 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
then r2 in { g2 where g2 is Real : ( r < g2 & g2 < (max (r1,r2)) + 1 ) } by A7;
then A8: r2 in ].r,((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r2 in dom f by A4, XBOOLE_0:def_4;
then A9: r2 in ].r,((max (r1,r2)) + 1).[ /\ (dom f) by A8, XBOOLE_0:def_4;
r1 in right_open_halfline r by A3, XBOOLE_0:def_4;
then r1 in { g where g is Real : r < g } by XXREAL_1:230;
then A10: ex g1 being Real st
( g1 = r1 & r < g1 ) ;
r1 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
then r1 in { g1 where g1 is Real : ( r < g1 & g1 < (max (r1,r2)) + 1 ) } by A10;
then A11: r1 in ].r,((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r1 in dom f by A3, XBOOLE_0:def_4;
then A12: r1 in ].r,((max (r1,r2)) + 1).[ /\ (dom f) by A11, XBOOLE_0:def_4;
].r,((max (r1,r2)) + 1).[ c= dom f by A1, A5, XBOOLE_1:1;
then f | ].r,((max (r1,r2)) + 1).[ is V8() by A6, Th25;
hence f . r1 = f . r2 by A12, A9, PARTFUN2:58; ::_thesis: verum
end;
hence ( f is_differentiable_on right_open_halfline r & f | (right_open_halfline r) is V8() ) by A1, A2, Th24, PARTFUN2:58; ::_thesis: verum
end;
theorem :: FDIFF_2:28
for f being PartFunc of REAL,REAL st f is total & ( for r1, r2 being Real holds abs ((f . r1) - (f . r2)) <= (r1 - r2) ^2 ) holds
( f is_differentiable_on [#] REAL & f | ([#] REAL) is V8() )
proof
let f be PartFunc of REAL,REAL; ::_thesis: ( f is total & ( for r1, r2 being Real holds abs ((f . r1) - (f . r2)) <= (r1 - r2) ^2 ) implies ( f is_differentiable_on [#] REAL & f | ([#] REAL) is V8() ) )
assume that
A1: f is total and
A2: for r1, r2 being Real holds abs ((f . r1) - (f . r2)) <= (r1 - r2) ^2 ; ::_thesis: ( f is_differentiable_on [#] REAL & f | ([#] REAL) is V8() )
A3: dom f = [#] REAL by A1, PARTFUN1:def_2;
A4: now__::_thesis:_for_r1,_r2_being_Real_st_r1_in_([#]_REAL)_/\_(dom_f)_&_r2_in_([#]_REAL)_/\_(dom_f)_holds_
f_._r1_=_f_._r2
let r1, r2 be Real; ::_thesis: ( r1 in ([#] REAL) /\ (dom f) & r2 in ([#] REAL) /\ (dom f) implies f . r1 = f . r2 )
assume that
A5: r1 in ([#] REAL) /\ (dom f) and
A6: r2 in ([#] REAL) /\ (dom f) ; ::_thesis: f . r1 = f . r2
set rx = max (r1,r2);
set rn = min (r1,r2);
A7: r1 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
A8: r2 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
(min (r1,r2)) - 1 < r2 - 0 by XREAL_1:15, XXREAL_0:17;
then r2 in { g2 where g2 is Real : ( (min (r1,r2)) - 1 < g2 & g2 < (max (r1,r2)) + 1 ) } by A8;
then A9: r2 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r2 in dom f by A6, XBOOLE_0:def_4;
then A10: r2 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ /\ (dom f) by A9, XBOOLE_0:def_4;
(min (r1,r2)) - 1 < r1 - 0 by XREAL_1:15, XXREAL_0:17;
then r1 in { g1 where g1 is Real : ( (min (r1,r2)) - 1 < g1 & g1 < (max (r1,r2)) + 1 ) } by A7;
then A11: r1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r1 in dom f by A5, XBOOLE_0:def_4;
then A12: r1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ /\ (dom f) by A11, XBOOLE_0:def_4;
for g1, g2 being Real st g1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ & g2 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ holds
abs ((f . g1) - (f . g2)) <= (g1 - g2) ^2 by A2;
then f | ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ is V8() by A3, Th25;
hence f . r1 = f . r2 by A12, A10, PARTFUN2:58; ::_thesis: verum
end;
for r1, r2 being Real st r1 in [#] REAL & r2 in [#] REAL holds
abs ((f . r1) - (f . r2)) <= (r1 - r2) ^2 by A2;
hence ( f is_differentiable_on [#] REAL & f | ([#] REAL) is V8() ) by A3, A4, Th24, PARTFUN2:58; ::_thesis: verum
end;
theorem Th29: :: FDIFF_2:29
for r being Real
for f being PartFunc of REAL,REAL st left_open_halfline r c= dom f & f is_differentiable_on left_open_halfline r & ( for x0 being Real st x0 in left_open_halfline r holds
0 < diff (f,x0) ) holds
( f | (left_open_halfline r) is increasing & f | (left_open_halfline r) is one-to-one )
proof
let r be Real; ::_thesis: for f being PartFunc of REAL,REAL st left_open_halfline r c= dom f & f is_differentiable_on left_open_halfline r & ( for x0 being Real st x0 in left_open_halfline r holds
0 < diff (f,x0) ) holds
( f | (left_open_halfline r) is increasing & f | (left_open_halfline r) is one-to-one )
let f be PartFunc of REAL,REAL; ::_thesis: ( left_open_halfline r c= dom f & f is_differentiable_on left_open_halfline r & ( for x0 being Real st x0 in left_open_halfline r holds
0 < diff (f,x0) ) implies ( f | (left_open_halfline r) is increasing & f | (left_open_halfline r) is one-to-one ) )
assume A1: left_open_halfline r c= dom f ; ::_thesis: ( not f is_differentiable_on left_open_halfline r or ex x0 being Real st
( x0 in left_open_halfline r & not 0 < diff (f,x0) ) or ( f | (left_open_halfline r) is increasing & f | (left_open_halfline r) is one-to-one ) )
assume that
A2: f is_differentiable_on left_open_halfline r and
A3: for x0 being Real st x0 in left_open_halfline r holds
0 < diff (f,x0) ; ::_thesis: ( f | (left_open_halfline r) is increasing & f | (left_open_halfline r) is one-to-one )
now__::_thesis:_for_r1,_r2_being_Real_st_r1_in_(left_open_halfline_r)_/\_(dom_f)_&_r2_in_(left_open_halfline_r)_/\_(dom_f)_&_r1_<_r2_holds_
f_._r1_<_f_._r2
let r1, r2 be Real; ::_thesis: ( r1 in (left_open_halfline r) /\ (dom f) & r2 in (left_open_halfline r) /\ (dom f) & r1 < r2 implies f . r1 < f . r2 )
assume that
A4: r1 in (left_open_halfline r) /\ (dom f) and
A5: r2 in (left_open_halfline r) /\ (dom f) and
A6: r1 < r2 ; ::_thesis: f . r1 < f . r2
set rr = min (r1,r2);
A7: (min (r1,r2)) - 1 < r2 - 0 by XREAL_1:15, XXREAL_0:17;
r2 in left_open_halfline r by A5, XBOOLE_0:def_4;
then r2 in { p where p is Real : p < r } by XXREAL_1:229;
then ex g2 being Real st
( g2 = r2 & g2 < r ) ;
then r2 in { g2 where g2 is Real : ( (min (r1,r2)) - 1 < g2 & g2 < r ) } by A7;
then A8: r2 in ].((min (r1,r2)) - 1),r.[ by RCOMP_1:def_2;
r2 in dom f by A5, XBOOLE_0:def_4;
then A9: r2 in ].((min (r1,r2)) - 1),r.[ /\ (dom f) by A8, XBOOLE_0:def_4;
A10: f is_differentiable_on ].((min (r1,r2)) - 1),r.[ by A2, FDIFF_1:26, XXREAL_1:263;
A11: ].((min (r1,r2)) - 1),r.[ c= left_open_halfline r by XXREAL_1:263;
then for g1 being Real st g1 in ].((min (r1,r2)) - 1),r.[ holds
0 < diff (f,g1) by A3;
then A12: f | ].((min (r1,r2)) - 1),r.[ is increasing by A1, A11, A10, ROLLE:9, XBOOLE_1:1;
A13: (min (r1,r2)) - 1 < r1 - 0 by XREAL_1:15, XXREAL_0:17;
r1 in left_open_halfline r by A4, XBOOLE_0:def_4;
then r1 in { g where g is Real : g < r } by XXREAL_1:229;
then ex g1 being Real st
( g1 = r1 & g1 < r ) ;
then r1 in { g1 where g1 is Real : ( (min (r1,r2)) - 1 < g1 & g1 < r ) } by A13;
then A14: r1 in ].((min (r1,r2)) - 1),r.[ by RCOMP_1:def_2;
r1 in dom f by A4, XBOOLE_0:def_4;
then r1 in ].((min (r1,r2)) - 1),r.[ /\ (dom f) by A14, XBOOLE_0:def_4;
hence f . r1 < f . r2 by A6, A12, A9, RFUNCT_2:20; ::_thesis: verum
end;
hence f | (left_open_halfline r) is increasing by RFUNCT_2:20; ::_thesis: f | (left_open_halfline r) is one-to-one
hence f | (left_open_halfline r) is one-to-one by FCONT_3:8; ::_thesis: verum
end;
theorem Th30: :: FDIFF_2:30
for r being Real
for f being PartFunc of REAL,REAL st left_open_halfline r c= dom f & f is_differentiable_on left_open_halfline r & ( for x0 being Real st x0 in left_open_halfline r holds
diff (f,x0) < 0 ) holds
( f | (left_open_halfline r) is decreasing & f | (left_open_halfline r) is one-to-one )
proof
let r be Real; ::_thesis: for f being PartFunc of REAL,REAL st left_open_halfline r c= dom f & f is_differentiable_on left_open_halfline r & ( for x0 being Real st x0 in left_open_halfline r holds
diff (f,x0) < 0 ) holds
( f | (left_open_halfline r) is decreasing & f | (left_open_halfline r) is one-to-one )
let f be PartFunc of REAL,REAL; ::_thesis: ( left_open_halfline r c= dom f & f is_differentiable_on left_open_halfline r & ( for x0 being Real st x0 in left_open_halfline r holds
diff (f,x0) < 0 ) implies ( f | (left_open_halfline r) is decreasing & f | (left_open_halfline r) is one-to-one ) )
assume A1: left_open_halfline r c= dom f ; ::_thesis: ( not f is_differentiable_on left_open_halfline r or ex x0 being Real st
( x0 in left_open_halfline r & not diff (f,x0) < 0 ) or ( f | (left_open_halfline r) is decreasing & f | (left_open_halfline r) is one-to-one ) )
assume that
A2: f is_differentiable_on left_open_halfline r and
A3: for x0 being Real st x0 in left_open_halfline r holds
diff (f,x0) < 0 ; ::_thesis: ( f | (left_open_halfline r) is decreasing & f | (left_open_halfline r) is one-to-one )
now__::_thesis:_for_r1,_r2_being_Real_st_r1_in_(left_open_halfline_r)_/\_(dom_f)_&_r2_in_(left_open_halfline_r)_/\_(dom_f)_&_r1_<_r2_holds_
f_._r2_<_f_._r1
let r1, r2 be Real; ::_thesis: ( r1 in (left_open_halfline r) /\ (dom f) & r2 in (left_open_halfline r) /\ (dom f) & r1 < r2 implies f . r2 < f . r1 )
assume that
A4: r1 in (left_open_halfline r) /\ (dom f) and
A5: r2 in (left_open_halfline r) /\ (dom f) and
A6: r1 < r2 ; ::_thesis: f . r2 < f . r1
set rr = min (r1,r2);
A7: (min (r1,r2)) - 1 < r2 - 0 by XREAL_1:15, XXREAL_0:17;
r2 in left_open_halfline r by A5, XBOOLE_0:def_4;
then r2 in { p where p is Real : p < r } by XXREAL_1:229;
then ex g2 being Real st
( g2 = r2 & g2 < r ) ;
then r2 in { g2 where g2 is Real : ( (min (r1,r2)) - 1 < g2 & g2 < r ) } by A7;
then A8: r2 in ].((min (r1,r2)) - 1),r.[ by RCOMP_1:def_2;
r2 in dom f by A5, XBOOLE_0:def_4;
then A9: r2 in ].((min (r1,r2)) - 1),r.[ /\ (dom f) by A8, XBOOLE_0:def_4;
A10: f is_differentiable_on ].((min (r1,r2)) - 1),r.[ by A2, FDIFF_1:26, XXREAL_1:263;
A11: ].((min (r1,r2)) - 1),r.[ c= left_open_halfline r by XXREAL_1:263;
then for g1 being Real st g1 in ].((min (r1,r2)) - 1),r.[ holds
diff (f,g1) < 0 by A3;
then A12: f | ].((min (r1,r2)) - 1),r.[ is decreasing by A1, A11, A10, ROLLE:10, XBOOLE_1:1;
A13: (min (r1,r2)) - 1 < r1 - 0 by XREAL_1:15, XXREAL_0:17;
r1 in left_open_halfline r by A4, XBOOLE_0:def_4;
then r1 in { g where g is Real : g < r } by XXREAL_1:229;
then ex g1 being Real st
( g1 = r1 & g1 < r ) ;
then r1 in { g1 where g1 is Real : ( (min (r1,r2)) - 1 < g1 & g1 < r ) } by A13;
then A14: r1 in ].((min (r1,r2)) - 1),r.[ by RCOMP_1:def_2;
r1 in dom f by A4, XBOOLE_0:def_4;
then r1 in ].((min (r1,r2)) - 1),r.[ /\ (dom f) by A14, XBOOLE_0:def_4;
hence f . r2 < f . r1 by A6, A12, A9, RFUNCT_2:21; ::_thesis: verum
end;
hence f | (left_open_halfline r) is decreasing by RFUNCT_2:21; ::_thesis: f | (left_open_halfline r) is one-to-one
hence f | (left_open_halfline r) is one-to-one by FCONT_3:8; ::_thesis: verum
end;
theorem :: FDIFF_2:31
for r being Real
for f being PartFunc of REAL,REAL st left_open_halfline r c= dom f & f is_differentiable_on left_open_halfline r & ( for x0 being Real st x0 in left_open_halfline r holds
0 <= diff (f,x0) ) holds
f | (left_open_halfline r) is non-decreasing
proof
let r be Real; ::_thesis: for f being PartFunc of REAL,REAL st left_open_halfline r c= dom f & f is_differentiable_on left_open_halfline r & ( for x0 being Real st x0 in left_open_halfline r holds
0 <= diff (f,x0) ) holds
f | (left_open_halfline r) is non-decreasing
let f be PartFunc of REAL,REAL; ::_thesis: ( left_open_halfline r c= dom f & f is_differentiable_on left_open_halfline r & ( for x0 being Real st x0 in left_open_halfline r holds
0 <= diff (f,x0) ) implies f | (left_open_halfline r) is non-decreasing )
assume A1: left_open_halfline r c= dom f ; ::_thesis: ( not f is_differentiable_on left_open_halfline r or ex x0 being Real st
( x0 in left_open_halfline r & not 0 <= diff (f,x0) ) or f | (left_open_halfline r) is non-decreasing )
assume that
A2: f is_differentiable_on left_open_halfline r and
A3: for x0 being Real st x0 in left_open_halfline r holds
0 <= diff (f,x0) ; ::_thesis: f | (left_open_halfline r) is non-decreasing
now__::_thesis:_for_r1,_r2_being_Real_st_r1_in_(left_open_halfline_r)_/\_(dom_f)_&_r2_in_(left_open_halfline_r)_/\_(dom_f)_&_r1_<_r2_holds_
f_._r1_<=_f_._r2
let r1, r2 be Real; ::_thesis: ( r1 in (left_open_halfline r) /\ (dom f) & r2 in (left_open_halfline r) /\ (dom f) & r1 < r2 implies f . r1 <= f . r2 )
assume that
A4: r1 in (left_open_halfline r) /\ (dom f) and
A5: r2 in (left_open_halfline r) /\ (dom f) and
A6: r1 < r2 ; ::_thesis: f . r1 <= f . r2
set rr = min (r1,r2);
A7: (min (r1,r2)) - 1 < r2 - 0 by XREAL_1:15, XXREAL_0:17;
r2 in left_open_halfline r by A5, XBOOLE_0:def_4;
then r2 in { p where p is Real : p < r } by XXREAL_1:229;
then ex g2 being Real st
( g2 = r2 & g2 < r ) ;
then r2 in { g2 where g2 is Real : ( (min (r1,r2)) - 1 < g2 & g2 < r ) } by A7;
then A8: r2 in ].((min (r1,r2)) - 1),r.[ by RCOMP_1:def_2;
r2 in dom f by A5, XBOOLE_0:def_4;
then A9: r2 in ].((min (r1,r2)) - 1),r.[ /\ (dom f) by A8, XBOOLE_0:def_4;
A10: f is_differentiable_on ].((min (r1,r2)) - 1),r.[ by A2, FDIFF_1:26, XXREAL_1:263;
A11: ].((min (r1,r2)) - 1),r.[ c= left_open_halfline r by XXREAL_1:263;
then for g1 being Real st g1 in ].((min (r1,r2)) - 1),r.[ holds
0 <= diff (f,g1) by A3;
then A12: f | ].((min (r1,r2)) - 1),r.[ is non-decreasing by A1, A11, A10, ROLLE:11, XBOOLE_1:1;
A13: (min (r1,r2)) - 1 < r1 - 0 by XREAL_1:15, XXREAL_0:17;
r1 in left_open_halfline r by A4, XBOOLE_0:def_4;
then r1 in { g where g is Real : g < r } by XXREAL_1:229;
then ex g1 being Real st
( g1 = r1 & g1 < r ) ;
then r1 in { g1 where g1 is Real : ( (min (r1,r2)) - 1 < g1 & g1 < r ) } by A13;
then A14: r1 in ].((min (r1,r2)) - 1),r.[ by RCOMP_1:def_2;
r1 in dom f by A4, XBOOLE_0:def_4;
then r1 in ].((min (r1,r2)) - 1),r.[ /\ (dom f) by A14, XBOOLE_0:def_4;
hence f . r1 <= f . r2 by A6, A12, A9, RFUNCT_2:22; ::_thesis: verum
end;
hence f | (left_open_halfline r) is non-decreasing by RFUNCT_2:22; ::_thesis: verum
end;
theorem :: FDIFF_2:32
for r being Real
for f being PartFunc of REAL,REAL st left_open_halfline r c= dom f & f is_differentiable_on left_open_halfline r & ( for x0 being Real st x0 in left_open_halfline r holds
diff (f,x0) <= 0 ) holds
f | (left_open_halfline r) is non-increasing
proof
let r be Real; ::_thesis: for f being PartFunc of REAL,REAL st left_open_halfline r c= dom f & f is_differentiable_on left_open_halfline r & ( for x0 being Real st x0 in left_open_halfline r holds
diff (f,x0) <= 0 ) holds
f | (left_open_halfline r) is non-increasing
let f be PartFunc of REAL,REAL; ::_thesis: ( left_open_halfline r c= dom f & f is_differentiable_on left_open_halfline r & ( for x0 being Real st x0 in left_open_halfline r holds
diff (f,x0) <= 0 ) implies f | (left_open_halfline r) is non-increasing )
assume A1: left_open_halfline r c= dom f ; ::_thesis: ( not f is_differentiable_on left_open_halfline r or ex x0 being Real st
( x0 in left_open_halfline r & not diff (f,x0) <= 0 ) or f | (left_open_halfline r) is non-increasing )
assume that
A2: f is_differentiable_on left_open_halfline r and
A3: for x0 being Real st x0 in left_open_halfline r holds
diff (f,x0) <= 0 ; ::_thesis: f | (left_open_halfline r) is non-increasing
now__::_thesis:_for_r1,_r2_being_Real_st_r1_in_(left_open_halfline_r)_/\_(dom_f)_&_r2_in_(left_open_halfline_r)_/\_(dom_f)_&_r1_<_r2_holds_
f_._r2_<=_f_._r1
let r1, r2 be Real; ::_thesis: ( r1 in (left_open_halfline r) /\ (dom f) & r2 in (left_open_halfline r) /\ (dom f) & r1 < r2 implies f . r2 <= f . r1 )
assume that
A4: r1 in (left_open_halfline r) /\ (dom f) and
A5: r2 in (left_open_halfline r) /\ (dom f) and
A6: r1 < r2 ; ::_thesis: f . r2 <= f . r1
set rr = min (r1,r2);
A7: (min (r1,r2)) - 1 < r2 - 0 by XREAL_1:15, XXREAL_0:17;
r2 in left_open_halfline r by A5, XBOOLE_0:def_4;
then r2 in { p where p is Real : p < r } by XXREAL_1:229;
then ex g2 being Real st
( g2 = r2 & g2 < r ) ;
then r2 in { g2 where g2 is Real : ( (min (r1,r2)) - 1 < g2 & g2 < r ) } by A7;
then A8: r2 in ].((min (r1,r2)) - 1),r.[ by RCOMP_1:def_2;
r2 in dom f by A5, XBOOLE_0:def_4;
then A9: r2 in ].((min (r1,r2)) - 1),r.[ /\ (dom f) by A8, XBOOLE_0:def_4;
A10: f is_differentiable_on ].((min (r1,r2)) - 1),r.[ by A2, FDIFF_1:26, XXREAL_1:263;
A11: ].((min (r1,r2)) - 1),r.[ c= left_open_halfline r by XXREAL_1:263;
then for g1 being Real st g1 in ].((min (r1,r2)) - 1),r.[ holds
diff (f,g1) <= 0 by A3;
then A12: f | ].((min (r1,r2)) - 1),r.[ is non-increasing by A1, A11, A10, ROLLE:12, XBOOLE_1:1;
A13: (min (r1,r2)) - 1 < r1 - 0 by XREAL_1:15, XXREAL_0:17;
r1 in left_open_halfline r by A4, XBOOLE_0:def_4;
then r1 in { g where g is Real : g < r } by XXREAL_1:229;
then ex g1 being Real st
( g1 = r1 & g1 < r ) ;
then r1 in { g1 where g1 is Real : ( (min (r1,r2)) - 1 < g1 & g1 < r ) } by A13;
then A14: r1 in ].((min (r1,r2)) - 1),r.[ by RCOMP_1:def_2;
r1 in dom f by A4, XBOOLE_0:def_4;
then r1 in ].((min (r1,r2)) - 1),r.[ /\ (dom f) by A14, XBOOLE_0:def_4;
hence f . r2 <= f . r1 by A6, A12, A9, RFUNCT_2:23; ::_thesis: verum
end;
hence f | (left_open_halfline r) is non-increasing by RFUNCT_2:23; ::_thesis: verum
end;
theorem Th33: :: FDIFF_2:33
for r being Real
for f being PartFunc of REAL,REAL st right_open_halfline r c= dom f & f is_differentiable_on right_open_halfline r & ( for x0 being Real st x0 in right_open_halfline r holds
0 < diff (f,x0) ) holds
( f | (right_open_halfline r) is increasing & f | (right_open_halfline r) is one-to-one )
proof
let r be Real; ::_thesis: for f being PartFunc of REAL,REAL st right_open_halfline r c= dom f & f is_differentiable_on right_open_halfline r & ( for x0 being Real st x0 in right_open_halfline r holds
0 < diff (f,x0) ) holds
( f | (right_open_halfline r) is increasing & f | (right_open_halfline r) is one-to-one )
let f be PartFunc of REAL,REAL; ::_thesis: ( right_open_halfline r c= dom f & f is_differentiable_on right_open_halfline r & ( for x0 being Real st x0 in right_open_halfline r holds
0 < diff (f,x0) ) implies ( f | (right_open_halfline r) is increasing & f | (right_open_halfline r) is one-to-one ) )
assume A1: right_open_halfline r c= dom f ; ::_thesis: ( not f is_differentiable_on right_open_halfline r or ex x0 being Real st
( x0 in right_open_halfline r & not 0 < diff (f,x0) ) or ( f | (right_open_halfline r) is increasing & f | (right_open_halfline r) is one-to-one ) )
assume that
A2: f is_differentiable_on right_open_halfline r and
A3: for x0 being Real st x0 in right_open_halfline r holds
0 < diff (f,x0) ; ::_thesis: ( f | (right_open_halfline r) is increasing & f | (right_open_halfline r) is one-to-one )
now__::_thesis:_for_r1,_r2_being_Real_st_r1_in_(right_open_halfline_r)_/\_(dom_f)_&_r2_in_(right_open_halfline_r)_/\_(dom_f)_&_r1_<_r2_holds_
f_._r1_<_f_._r2
let r1, r2 be Real; ::_thesis: ( r1 in (right_open_halfline r) /\ (dom f) & r2 in (right_open_halfline r) /\ (dom f) & r1 < r2 implies f . r1 < f . r2 )
assume that
A4: r1 in (right_open_halfline r) /\ (dom f) and
A5: r2 in (right_open_halfline r) /\ (dom f) and
A6: r1 < r2 ; ::_thesis: f . r1 < f . r2
set rr = max (r1,r2);
A7: r2 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
r2 in right_open_halfline r by A5, XBOOLE_0:def_4;
then r2 in { p where p is Real : r < p } by XXREAL_1:230;
then ex g2 being Real st
( g2 = r2 & r < g2 ) ;
then r2 in { g2 where g2 is Real : ( r < g2 & g2 < (max (r1,r2)) + 1 ) } by A7;
then A8: r2 in ].r,((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r2 in dom f by A5, XBOOLE_0:def_4;
then A9: r2 in ].r,((max (r1,r2)) + 1).[ /\ (dom f) by A8, XBOOLE_0:def_4;
A10: f is_differentiable_on ].r,((max (r1,r2)) + 1).[ by A2, FDIFF_1:26, XXREAL_1:247;
A11: ].r,((max (r1,r2)) + 1).[ c= right_open_halfline r by XXREAL_1:247;
then for g1 being Real st g1 in ].r,((max (r1,r2)) + 1).[ holds
0 < diff (f,g1) by A3;
then A12: f | ].r,((max (r1,r2)) + 1).[ is increasing by A1, A11, A10, ROLLE:9, XBOOLE_1:1;
A13: r1 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
r1 in right_open_halfline r by A4, XBOOLE_0:def_4;
then r1 in { g where g is Real : r < g } by XXREAL_1:230;
then ex g1 being Real st
( g1 = r1 & r < g1 ) ;
then r1 in { g1 where g1 is Real : ( r < g1 & g1 < (max (r1,r2)) + 1 ) } by A13;
then A14: r1 in ].r,((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r1 in dom f by A4, XBOOLE_0:def_4;
then r1 in ].r,((max (r1,r2)) + 1).[ /\ (dom f) by A14, XBOOLE_0:def_4;
hence f . r1 < f . r2 by A6, A12, A9, RFUNCT_2:20; ::_thesis: verum
end;
hence f | (right_open_halfline r) is increasing by RFUNCT_2:20; ::_thesis: f | (right_open_halfline r) is one-to-one
hence f | (right_open_halfline r) is one-to-one by FCONT_3:8; ::_thesis: verum
end;
theorem Th34: :: FDIFF_2:34
for r being Real
for f being PartFunc of REAL,REAL st right_open_halfline r c= dom f & f is_differentiable_on right_open_halfline r & ( for x0 being Real st x0 in right_open_halfline r holds
diff (f,x0) < 0 ) holds
( f | (right_open_halfline r) is decreasing & f | (right_open_halfline r) is one-to-one )
proof
let r be Real; ::_thesis: for f being PartFunc of REAL,REAL st right_open_halfline r c= dom f & f is_differentiable_on right_open_halfline r & ( for x0 being Real st x0 in right_open_halfline r holds
diff (f,x0) < 0 ) holds
( f | (right_open_halfline r) is decreasing & f | (right_open_halfline r) is one-to-one )
let f be PartFunc of REAL,REAL; ::_thesis: ( right_open_halfline r c= dom f & f is_differentiable_on right_open_halfline r & ( for x0 being Real st x0 in right_open_halfline r holds
diff (f,x0) < 0 ) implies ( f | (right_open_halfline r) is decreasing & f | (right_open_halfline r) is one-to-one ) )
assume A1: right_open_halfline r c= dom f ; ::_thesis: ( not f is_differentiable_on right_open_halfline r or ex x0 being Real st
( x0 in right_open_halfline r & not diff (f,x0) < 0 ) or ( f | (right_open_halfline r) is decreasing & f | (right_open_halfline r) is one-to-one ) )
assume that
A2: f is_differentiable_on right_open_halfline r and
A3: for x0 being Real st x0 in right_open_halfline r holds
diff (f,x0) < 0 ; ::_thesis: ( f | (right_open_halfline r) is decreasing & f | (right_open_halfline r) is one-to-one )
now__::_thesis:_for_r1,_r2_being_Real_st_r1_in_(right_open_halfline_r)_/\_(dom_f)_&_r2_in_(right_open_halfline_r)_/\_(dom_f)_&_r1_<_r2_holds_
f_._r2_<_f_._r1
let r1, r2 be Real; ::_thesis: ( r1 in (right_open_halfline r) /\ (dom f) & r2 in (right_open_halfline r) /\ (dom f) & r1 < r2 implies f . r2 < f . r1 )
assume that
A4: r1 in (right_open_halfline r) /\ (dom f) and
A5: r2 in (right_open_halfline r) /\ (dom f) and
A6: r1 < r2 ; ::_thesis: f . r2 < f . r1
set rr = max (r1,r2);
A7: r2 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
r2 in right_open_halfline r by A5, XBOOLE_0:def_4;
then r2 in { p where p is Real : r < p } by XXREAL_1:230;
then ex g2 being Real st
( g2 = r2 & r < g2 ) ;
then r2 in { g2 where g2 is Real : ( r < g2 & g2 < (max (r1,r2)) + 1 ) } by A7;
then A8: r2 in ].r,((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r2 in dom f by A5, XBOOLE_0:def_4;
then A9: r2 in ].r,((max (r1,r2)) + 1).[ /\ (dom f) by A8, XBOOLE_0:def_4;
A10: f is_differentiable_on ].r,((max (r1,r2)) + 1).[ by A2, FDIFF_1:26, XXREAL_1:247;
A11: ].r,((max (r1,r2)) + 1).[ c= right_open_halfline r by XXREAL_1:247;
then for g1 being Real st g1 in ].r,((max (r1,r2)) + 1).[ holds
diff (f,g1) < 0 by A3;
then A12: f | ].r,((max (r1,r2)) + 1).[ is decreasing by A1, A11, A10, ROLLE:10, XBOOLE_1:1;
A13: r1 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
r1 in right_open_halfline r by A4, XBOOLE_0:def_4;
then r1 in { g where g is Real : r < g } by XXREAL_1:230;
then ex g1 being Real st
( g1 = r1 & r < g1 ) ;
then r1 in { g1 where g1 is Real : ( r < g1 & g1 < (max (r1,r2)) + 1 ) } by A13;
then A14: r1 in ].r,((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r1 in dom f by A4, XBOOLE_0:def_4;
then r1 in ].r,((max (r1,r2)) + 1).[ /\ (dom f) by A14, XBOOLE_0:def_4;
hence f . r2 < f . r1 by A6, A12, A9, RFUNCT_2:21; ::_thesis: verum
end;
hence f | (right_open_halfline r) is decreasing by RFUNCT_2:21; ::_thesis: f | (right_open_halfline r) is one-to-one
hence f | (right_open_halfline r) is one-to-one by FCONT_3:8; ::_thesis: verum
end;
theorem :: FDIFF_2:35
for r being Real
for f being PartFunc of REAL,REAL st right_open_halfline r c= dom f & f is_differentiable_on right_open_halfline r & ( for x0 being Real st x0 in right_open_halfline r holds
0 <= diff (f,x0) ) holds
f | (right_open_halfline r) is non-decreasing
proof
let r be Real; ::_thesis: for f being PartFunc of REAL,REAL st right_open_halfline r c= dom f & f is_differentiable_on right_open_halfline r & ( for x0 being Real st x0 in right_open_halfline r holds
0 <= diff (f,x0) ) holds
f | (right_open_halfline r) is non-decreasing
let f be PartFunc of REAL,REAL; ::_thesis: ( right_open_halfline r c= dom f & f is_differentiable_on right_open_halfline r & ( for x0 being Real st x0 in right_open_halfline r holds
0 <= diff (f,x0) ) implies f | (right_open_halfline r) is non-decreasing )
assume A1: right_open_halfline r c= dom f ; ::_thesis: ( not f is_differentiable_on right_open_halfline r or ex x0 being Real st
( x0 in right_open_halfline r & not 0 <= diff (f,x0) ) or f | (right_open_halfline r) is non-decreasing )
assume that
A2: f is_differentiable_on right_open_halfline r and
A3: for x0 being Real st x0 in right_open_halfline r holds
0 <= diff (f,x0) ; ::_thesis: f | (right_open_halfline r) is non-decreasing
now__::_thesis:_for_r1,_r2_being_Real_st_r1_in_(right_open_halfline_r)_/\_(dom_f)_&_r2_in_(right_open_halfline_r)_/\_(dom_f)_&_r1_<_r2_holds_
f_._r1_<=_f_._r2
let r1, r2 be Real; ::_thesis: ( r1 in (right_open_halfline r) /\ (dom f) & r2 in (right_open_halfline r) /\ (dom f) & r1 < r2 implies f . r1 <= f . r2 )
assume that
A4: r1 in (right_open_halfline r) /\ (dom f) and
A5: r2 in (right_open_halfline r) /\ (dom f) and
A6: r1 < r2 ; ::_thesis: f . r1 <= f . r2
set rr = max (r1,r2);
A7: r2 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
r2 in right_open_halfline r by A5, XBOOLE_0:def_4;
then r2 in { p where p is Real : r < p } by XXREAL_1:230;
then ex g2 being Real st
( g2 = r2 & r < g2 ) ;
then r2 in { g2 where g2 is Real : ( r < g2 & g2 < (max (r1,r2)) + 1 ) } by A7;
then A8: r2 in ].r,((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r2 in dom f by A5, XBOOLE_0:def_4;
then A9: r2 in ].r,((max (r1,r2)) + 1).[ /\ (dom f) by A8, XBOOLE_0:def_4;
A10: f is_differentiable_on ].r,((max (r1,r2)) + 1).[ by A2, FDIFF_1:26, XXREAL_1:247;
A11: ].r,((max (r1,r2)) + 1).[ c= right_open_halfline r by XXREAL_1:247;
then for g1 being Real st g1 in ].r,((max (r1,r2)) + 1).[ holds
0 <= diff (f,g1) by A3;
then A12: f | ].r,((max (r1,r2)) + 1).[ is non-decreasing by A1, A11, A10, ROLLE:11, XBOOLE_1:1;
A13: r1 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
r1 in right_open_halfline r by A4, XBOOLE_0:def_4;
then r1 in { g where g is Real : r < g } by XXREAL_1:230;
then ex g1 being Real st
( g1 = r1 & r < g1 ) ;
then r1 in { g1 where g1 is Real : ( r < g1 & g1 < (max (r1,r2)) + 1 ) } by A13;
then A14: r1 in ].r,((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r1 in dom f by A4, XBOOLE_0:def_4;
then r1 in ].r,((max (r1,r2)) + 1).[ /\ (dom f) by A14, XBOOLE_0:def_4;
hence f . r1 <= f . r2 by A6, A12, A9, RFUNCT_2:22; ::_thesis: verum
end;
hence f | (right_open_halfline r) is non-decreasing by RFUNCT_2:22; ::_thesis: verum
end;
theorem :: FDIFF_2:36
for r being Real
for f being PartFunc of REAL,REAL st right_open_halfline r c= dom f & f is_differentiable_on right_open_halfline r & ( for x0 being Real st x0 in right_open_halfline r holds
diff (f,x0) <= 0 ) holds
f | (right_open_halfline r) is non-increasing
proof
let r be Real; ::_thesis: for f being PartFunc of REAL,REAL st right_open_halfline r c= dom f & f is_differentiable_on right_open_halfline r & ( for x0 being Real st x0 in right_open_halfline r holds
diff (f,x0) <= 0 ) holds
f | (right_open_halfline r) is non-increasing
let f be PartFunc of REAL,REAL; ::_thesis: ( right_open_halfline r c= dom f & f is_differentiable_on right_open_halfline r & ( for x0 being Real st x0 in right_open_halfline r holds
diff (f,x0) <= 0 ) implies f | (right_open_halfline r) is non-increasing )
assume A1: right_open_halfline r c= dom f ; ::_thesis: ( not f is_differentiable_on right_open_halfline r or ex x0 being Real st
( x0 in right_open_halfline r & not diff (f,x0) <= 0 ) or f | (right_open_halfline r) is non-increasing )
assume that
A2: f is_differentiable_on right_open_halfline r and
A3: for x0 being Real st x0 in right_open_halfline r holds
diff (f,x0) <= 0 ; ::_thesis: f | (right_open_halfline r) is non-increasing
now__::_thesis:_for_r1,_r2_being_Real_st_r1_in_(right_open_halfline_r)_/\_(dom_f)_&_r2_in_(right_open_halfline_r)_/\_(dom_f)_&_r1_<_r2_holds_
f_._r2_<=_f_._r1
let r1, r2 be Real; ::_thesis: ( r1 in (right_open_halfline r) /\ (dom f) & r2 in (right_open_halfline r) /\ (dom f) & r1 < r2 implies f . r2 <= f . r1 )
assume that
A4: r1 in (right_open_halfline r) /\ (dom f) and
A5: r2 in (right_open_halfline r) /\ (dom f) and
A6: r1 < r2 ; ::_thesis: f . r2 <= f . r1
set rr = max (r1,r2);
A7: r2 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
r2 in right_open_halfline r by A5, XBOOLE_0:def_4;
then r2 in { p where p is Real : r < p } by XXREAL_1:230;
then ex g2 being Real st
( g2 = r2 & r < g2 ) ;
then r2 in { g2 where g2 is Real : ( r < g2 & g2 < (max (r1,r2)) + 1 ) } by A7;
then A8: r2 in ].r,((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r2 in dom f by A5, XBOOLE_0:def_4;
then A9: r2 in ].r,((max (r1,r2)) + 1).[ /\ (dom f) by A8, XBOOLE_0:def_4;
A10: f is_differentiable_on ].r,((max (r1,r2)) + 1).[ by A2, FDIFF_1:26, XXREAL_1:247;
A11: ].r,((max (r1,r2)) + 1).[ c= right_open_halfline r by XXREAL_1:247;
then for g1 being Real st g1 in ].r,((max (r1,r2)) + 1).[ holds
diff (f,g1) <= 0 by A3;
then A12: f | ].r,((max (r1,r2)) + 1).[ is non-increasing by A1, A11, A10, ROLLE:12, XBOOLE_1:1;
A13: r1 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
r1 in right_open_halfline r by A4, XBOOLE_0:def_4;
then r1 in { g where g is Real : r < g } by XXREAL_1:230;
then ex g1 being Real st
( g1 = r1 & r < g1 ) ;
then r1 in { g1 where g1 is Real : ( r < g1 & g1 < (max (r1,r2)) + 1 ) } by A13;
then A14: r1 in ].r,((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r1 in dom f by A4, XBOOLE_0:def_4;
then r1 in ].r,((max (r1,r2)) + 1).[ /\ (dom f) by A14, XBOOLE_0:def_4;
hence f . r2 <= f . r1 by A6, A12, A9, RFUNCT_2:23; ::_thesis: verum
end;
hence f | (right_open_halfline r) is non-increasing by RFUNCT_2:23; ::_thesis: verum
end;
theorem Th37: :: FDIFF_2:37
for f being PartFunc of REAL,REAL st [#] REAL c= dom f & f is_differentiable_on [#] REAL & ( for x0 being Real holds 0 < diff (f,x0) ) holds
( f | ([#] REAL) is increasing & f is one-to-one )
proof
let f be PartFunc of REAL,REAL; ::_thesis: ( [#] REAL c= dom f & f is_differentiable_on [#] REAL & ( for x0 being Real holds 0 < diff (f,x0) ) implies ( f | ([#] REAL) is increasing & f is one-to-one ) )
assume A1: [#] REAL c= dom f ; ::_thesis: ( not f is_differentiable_on [#] REAL or ex x0 being Real st not 0 < diff (f,x0) or ( f | ([#] REAL) is increasing & f is one-to-one ) )
assume that
A2: f is_differentiable_on [#] REAL and
A3: for x0 being Real holds 0 < diff (f,x0) ; ::_thesis: ( f | ([#] REAL) is increasing & f is one-to-one )
A4: now__::_thesis:_for_r1,_r2_being_Real_st_r1_in_([#]_REAL)_/\_(dom_f)_&_r2_in_([#]_REAL)_/\_(dom_f)_&_r1_<_r2_holds_
f_._r1_<_f_._r2
let r1, r2 be Real; ::_thesis: ( r1 in ([#] REAL) /\ (dom f) & r2 in ([#] REAL) /\ (dom f) & r1 < r2 implies f . r1 < f . r2 )
assume that
A5: r1 in ([#] REAL) /\ (dom f) and
A6: r2 in ([#] REAL) /\ (dom f) and
A7: r1 < r2 ; ::_thesis: f . r1 < f . r2
set rx = max (r1,r2);
set rn = min (r1,r2);
A8: r2 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
(min (r1,r2)) - 1 < r2 - 0 by XREAL_1:15, XXREAL_0:17;
then r2 in { g2 where g2 is Real : ( (min (r1,r2)) - 1 < g2 & g2 < (max (r1,r2)) + 1 ) } by A8;
then A9: r2 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r2 in dom f by A6, XBOOLE_0:def_4;
then A10: r2 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ /\ (dom f) by A9, XBOOLE_0:def_4;
A11: for g1 being Real st g1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ holds
0 < diff (f,g1) by A3;
f is_differentiable_on ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ by A2, FDIFF_1:26;
then A12: f | ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ is increasing by A1, A11, ROLLE:9, XBOOLE_1:1;
A13: r1 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
(min (r1,r2)) - 1 < r1 - 0 by XREAL_1:15, XXREAL_0:17;
then r1 in { g1 where g1 is Real : ( (min (r1,r2)) - 1 < g1 & g1 < (max (r1,r2)) + 1 ) } by A13;
then A14: r1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r1 in dom f by A5, XBOOLE_0:def_4;
then r1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ /\ (dom f) by A14, XBOOLE_0:def_4;
hence f . r1 < f . r2 by A7, A12, A10, RFUNCT_2:20; ::_thesis: verum
end;
hence f | ([#] REAL) is increasing by RFUNCT_2:20; ::_thesis: f is one-to-one
now__::_thesis:_for_r1,_r2_being_Real_holds_
(_not_r1_in_dom_f_or_not_r2_in_dom_f_or_not_f_._r1_=_f_._r2_or_not_r1_<>_r2_)
given r1, r2 being Real such that A15: r1 in dom f and
A16: r2 in dom f and
A17: f . r1 = f . r2 and
A18: r1 <> r2 ; ::_thesis: contradiction
A19: r2 in ([#] REAL) /\ (dom f) by A16, XBOOLE_0:def_4;
A20: r1 in ([#] REAL) /\ (dom f) by A15, XBOOLE_0:def_4;
now__::_thesis:_contradiction
percases ( r1 < r2 or r2 < r1 ) by A18, XXREAL_0:1;
suppose r1 < r2 ; ::_thesis: contradiction
hence contradiction by A4, A17, A20, A19; ::_thesis: verum
end;
suppose r2 < r1 ; ::_thesis: contradiction
hence contradiction by A4, A17, A20, A19; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence f is one-to-one by PARTFUN1:8; ::_thesis: verum
end;
theorem Th38: :: FDIFF_2:38
for f being PartFunc of REAL,REAL st [#] REAL c= dom f & f is_differentiable_on [#] REAL & ( for x0 being Real holds diff (f,x0) < 0 ) holds
( f | ([#] REAL) is decreasing & f is one-to-one )
proof
let f be PartFunc of REAL,REAL; ::_thesis: ( [#] REAL c= dom f & f is_differentiable_on [#] REAL & ( for x0 being Real holds diff (f,x0) < 0 ) implies ( f | ([#] REAL) is decreasing & f is one-to-one ) )
assume A1: [#] REAL c= dom f ; ::_thesis: ( not f is_differentiable_on [#] REAL or ex x0 being Real st not diff (f,x0) < 0 or ( f | ([#] REAL) is decreasing & f is one-to-one ) )
assume that
A2: f is_differentiable_on [#] REAL and
A3: for x0 being Real holds diff (f,x0) < 0 ; ::_thesis: ( f | ([#] REAL) is decreasing & f is one-to-one )
A4: now__::_thesis:_for_r1,_r2_being_Real_st_r1_in_([#]_REAL)_/\_(dom_f)_&_r2_in_([#]_REAL)_/\_(dom_f)_&_r1_<_r2_holds_
f_._r2_<_f_._r1
let r1, r2 be Real; ::_thesis: ( r1 in ([#] REAL) /\ (dom f) & r2 in ([#] REAL) /\ (dom f) & r1 < r2 implies f . r2 < f . r1 )
assume that
A5: r1 in ([#] REAL) /\ (dom f) and
A6: r2 in ([#] REAL) /\ (dom f) and
A7: r1 < r2 ; ::_thesis: f . r2 < f . r1
set rx = max (r1,r2);
set rn = min (r1,r2);
A8: r2 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
(min (r1,r2)) - 1 < r2 - 0 by XREAL_1:15, XXREAL_0:17;
then r2 in { g2 where g2 is Real : ( (min (r1,r2)) - 1 < g2 & g2 < (max (r1,r2)) + 1 ) } by A8;
then A9: r2 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r2 in dom f by A6, XBOOLE_0:def_4;
then A10: r2 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ /\ (dom f) by A9, XBOOLE_0:def_4;
A11: for g1 being Real st g1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ holds
diff (f,g1) < 0 by A3;
f is_differentiable_on ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ by A2, FDIFF_1:26;
then A12: f | ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ is decreasing by A1, A11, ROLLE:10, XBOOLE_1:1;
A13: r1 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
(min (r1,r2)) - 1 < r1 - 0 by XREAL_1:15, XXREAL_0:17;
then r1 in { g1 where g1 is Real : ( (min (r1,r2)) - 1 < g1 & g1 < (max (r1,r2)) + 1 ) } by A13;
then A14: r1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r1 in dom f by A5, XBOOLE_0:def_4;
then r1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ /\ (dom f) by A14, XBOOLE_0:def_4;
hence f . r2 < f . r1 by A7, A12, A10, RFUNCT_2:21; ::_thesis: verum
end;
hence f | ([#] REAL) is decreasing by RFUNCT_2:21; ::_thesis: f is one-to-one
now__::_thesis:_for_r1,_r2_being_Real_holds_
(_not_r1_in_dom_f_or_not_r2_in_dom_f_or_not_f_._r1_=_f_._r2_or_not_r1_<>_r2_)
given r1, r2 being Real such that A15: r1 in dom f and
A16: r2 in dom f and
A17: f . r1 = f . r2 and
A18: r1 <> r2 ; ::_thesis: contradiction
A19: r2 in ([#] REAL) /\ (dom f) by A16, XBOOLE_0:def_4;
A20: r1 in ([#] REAL) /\ (dom f) by A15, XBOOLE_0:def_4;
now__::_thesis:_contradiction
percases ( r1 < r2 or r2 < r1 ) by A18, XXREAL_0:1;
suppose r1 < r2 ; ::_thesis: contradiction
hence contradiction by A4, A17, A20, A19; ::_thesis: verum
end;
suppose r2 < r1 ; ::_thesis: contradiction
hence contradiction by A4, A17, A20, A19; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence f is one-to-one by PARTFUN1:8; ::_thesis: verum
end;
theorem :: FDIFF_2:39
for f being PartFunc of REAL,REAL st [#] REAL c= dom f & f is_differentiable_on [#] REAL & ( for x0 being Real holds 0 <= diff (f,x0) ) holds
f | ([#] REAL) is non-decreasing
proof
let f be PartFunc of REAL,REAL; ::_thesis: ( [#] REAL c= dom f & f is_differentiable_on [#] REAL & ( for x0 being Real holds 0 <= diff (f,x0) ) implies f | ([#] REAL) is non-decreasing )
assume A1: [#] REAL c= dom f ; ::_thesis: ( not f is_differentiable_on [#] REAL or ex x0 being Real st not 0 <= diff (f,x0) or f | ([#] REAL) is non-decreasing )
assume that
A2: f is_differentiable_on [#] REAL and
A3: for x0 being Real holds 0 <= diff (f,x0) ; ::_thesis: f | ([#] REAL) is non-decreasing
now__::_thesis:_for_r1,_r2_being_Real_st_r1_in_([#]_REAL)_/\_(dom_f)_&_r2_in_([#]_REAL)_/\_(dom_f)_&_r1_<_r2_holds_
f_._r1_<=_f_._r2
let r1, r2 be Real; ::_thesis: ( r1 in ([#] REAL) /\ (dom f) & r2 in ([#] REAL) /\ (dom f) & r1 < r2 implies f . r1 <= f . r2 )
assume that
A4: r1 in ([#] REAL) /\ (dom f) and
A5: r2 in ([#] REAL) /\ (dom f) and
A6: r1 < r2 ; ::_thesis: f . r1 <= f . r2
set rx = max (r1,r2);
set rn = min (r1,r2);
A7: r2 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
(min (r1,r2)) - 1 < r2 - 0 by XREAL_1:15, XXREAL_0:17;
then r2 in { g2 where g2 is Real : ( (min (r1,r2)) - 1 < g2 & g2 < (max (r1,r2)) + 1 ) } by A7;
then A8: r2 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r2 in dom f by A5, XBOOLE_0:def_4;
then A9: r2 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ /\ (dom f) by A8, XBOOLE_0:def_4;
A10: for g1 being Real st g1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ holds
0 <= diff (f,g1) by A3;
f is_differentiable_on ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ by A2, FDIFF_1:26;
then A11: f | ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ is non-decreasing by A1, A10, ROLLE:11, XBOOLE_1:1;
A12: r1 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
(min (r1,r2)) - 1 < r1 - 0 by XREAL_1:15, XXREAL_0:17;
then r1 in { g1 where g1 is Real : ( (min (r1,r2)) - 1 < g1 & g1 < (max (r1,r2)) + 1 ) } by A12;
then A13: r1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r1 in dom f by A4, XBOOLE_0:def_4;
then r1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ /\ (dom f) by A13, XBOOLE_0:def_4;
hence f . r1 <= f . r2 by A6, A11, A9, RFUNCT_2:22; ::_thesis: verum
end;
hence f | ([#] REAL) is non-decreasing by RFUNCT_2:22; ::_thesis: verum
end;
theorem :: FDIFF_2:40
for f being PartFunc of REAL,REAL st [#] REAL c= dom f & f is_differentiable_on [#] REAL & ( for x0 being Real holds diff (f,x0) <= 0 ) holds
f | ([#] REAL) is non-increasing
proof
let f be PartFunc of REAL,REAL; ::_thesis: ( [#] REAL c= dom f & f is_differentiable_on [#] REAL & ( for x0 being Real holds diff (f,x0) <= 0 ) implies f | ([#] REAL) is non-increasing )
assume A1: [#] REAL c= dom f ; ::_thesis: ( not f is_differentiable_on [#] REAL or ex x0 being Real st not diff (f,x0) <= 0 or f | ([#] REAL) is non-increasing )
assume that
A2: f is_differentiable_on [#] REAL and
A3: for x0 being Real holds diff (f,x0) <= 0 ; ::_thesis: f | ([#] REAL) is non-increasing
now__::_thesis:_for_r1,_r2_being_Real_st_r1_in_([#]_REAL)_/\_(dom_f)_&_r2_in_([#]_REAL)_/\_(dom_f)_&_r1_<_r2_holds_
f_._r2_<=_f_._r1
let r1, r2 be Real; ::_thesis: ( r1 in ([#] REAL) /\ (dom f) & r2 in ([#] REAL) /\ (dom f) & r1 < r2 implies f . r2 <= f . r1 )
assume that
A4: r1 in ([#] REAL) /\ (dom f) and
A5: r2 in ([#] REAL) /\ (dom f) and
A6: r1 < r2 ; ::_thesis: f . r2 <= f . r1
set rx = max (r1,r2);
set rn = min (r1,r2);
A7: r2 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
(min (r1,r2)) - 1 < r2 - 0 by XREAL_1:15, XXREAL_0:17;
then r2 in { g2 where g2 is Real : ( (min (r1,r2)) - 1 < g2 & g2 < (max (r1,r2)) + 1 ) } by A7;
then A8: r2 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r2 in dom f by A5, XBOOLE_0:def_4;
then A9: r2 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ /\ (dom f) by A8, XBOOLE_0:def_4;
A10: for g1 being Real st g1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ holds
diff (f,g1) <= 0 by A3;
f is_differentiable_on ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ by A2, FDIFF_1:26;
then A11: f | ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ is non-increasing by A1, A10, ROLLE:12, XBOOLE_1:1;
A12: r1 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
(min (r1,r2)) - 1 < r1 - 0 by XREAL_1:15, XXREAL_0:17;
then r1 in { g1 where g1 is Real : ( (min (r1,r2)) - 1 < g1 & g1 < (max (r1,r2)) + 1 ) } by A12;
then A13: r1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ by RCOMP_1:def_2;
r1 in dom f by A4, XBOOLE_0:def_4;
then r1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ /\ (dom f) by A13, XBOOLE_0:def_4;
hence f . r2 <= f . r1 by A6, A11, A9, RFUNCT_2:23; ::_thesis: verum
end;
hence f | ([#] REAL) is non-increasing by RFUNCT_2:23; ::_thesis: verum
end;
theorem Th41: :: FDIFF_2:41
for p, g being Real
for f being PartFunc of REAL,REAL st ].p,g.[ c= dom f & f is_differentiable_on ].p,g.[ & ( for x0 being Real st x0 in ].p,g.[ holds
0 < diff (f,x0) or for x0 being Real st x0 in ].p,g.[ holds
diff (f,x0) < 0 ) holds
rng (f | ].p,g.[) is open
proof
let p, g be Real; ::_thesis: for f being PartFunc of REAL,REAL st ].p,g.[ c= dom f & f is_differentiable_on ].p,g.[ & ( for x0 being Real st x0 in ].p,g.[ holds
0 < diff (f,x0) or for x0 being Real st x0 in ].p,g.[ holds
diff (f,x0) < 0 ) holds
rng (f | ].p,g.[) is open
let f be PartFunc of REAL,REAL; ::_thesis: ( ].p,g.[ c= dom f & f is_differentiable_on ].p,g.[ & ( for x0 being Real st x0 in ].p,g.[ holds
0 < diff (f,x0) or for x0 being Real st x0 in ].p,g.[ holds
diff (f,x0) < 0 ) implies rng (f | ].p,g.[) is open )
assume A1: ].p,g.[ c= dom f ; ::_thesis: ( not f is_differentiable_on ].p,g.[ or ( ex x0 being Real st
( x0 in ].p,g.[ & not 0 < diff (f,x0) ) & ex x0 being Real st
( x0 in ].p,g.[ & not diff (f,x0) < 0 ) ) or rng (f | ].p,g.[) is open )
assume that
A2: f is_differentiable_on ].p,g.[ and
A3: ( for x0 being Real st x0 in ].p,g.[ holds
0 < diff (f,x0) or for x0 being Real st x0 in ].p,g.[ holds
diff (f,x0) < 0 ) ; ::_thesis: rng (f | ].p,g.[) is open
A4: f | ].p,g.[ is continuous by A2, FDIFF_1:25;
now__::_thesis:_rng_(f_|_].p,g.[)_is_open
percases ( for x0 being Real st x0 in ].p,g.[ holds
0 < diff (f,x0) or for x0 being Real st x0 in ].p,g.[ holds
diff (f,x0) < 0 ) by A3;
suppose for x0 being Real st x0 in ].p,g.[ holds
0 < diff (f,x0) ; ::_thesis: rng (f | ].p,g.[) is open
then f | ].p,g.[ is increasing by A1, A2, ROLLE:9;
hence rng (f | ].p,g.[) is open by A1, A4, FCONT_3:23; ::_thesis: verum
end;
suppose for x0 being Real st x0 in ].p,g.[ holds
diff (f,x0) < 0 ; ::_thesis: rng (f | ].p,g.[) is open
then f | ].p,g.[ is decreasing by A1, A2, ROLLE:10;
hence rng (f | ].p,g.[) is open by A1, A4, FCONT_3:23; ::_thesis: verum
end;
end;
end;
hence rng (f | ].p,g.[) is open ; ::_thesis: verum
end;
theorem Th42: :: FDIFF_2:42
for p being Real
for f being PartFunc of REAL,REAL st left_open_halfline p c= dom f & f is_differentiable_on left_open_halfline p & ( for x0 being Real st x0 in left_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in left_open_halfline p holds
diff (f,x0) < 0 ) holds
rng (f | (left_open_halfline p)) is open
proof
let p be Real; ::_thesis: for f being PartFunc of REAL,REAL st left_open_halfline p c= dom f & f is_differentiable_on left_open_halfline p & ( for x0 being Real st x0 in left_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in left_open_halfline p holds
diff (f,x0) < 0 ) holds
rng (f | (left_open_halfline p)) is open
let f be PartFunc of REAL,REAL; ::_thesis: ( left_open_halfline p c= dom f & f is_differentiable_on left_open_halfline p & ( for x0 being Real st x0 in left_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in left_open_halfline p holds
diff (f,x0) < 0 ) implies rng (f | (left_open_halfline p)) is open )
set L = left_open_halfline p;
assume A1: left_open_halfline p c= dom f ; ::_thesis: ( not f is_differentiable_on left_open_halfline p or ( ex x0 being Real st
( x0 in left_open_halfline p & not 0 < diff (f,x0) ) & ex x0 being Real st
( x0 in left_open_halfline p & not diff (f,x0) < 0 ) ) or rng (f | (left_open_halfline p)) is open )
assume that
A2: f is_differentiable_on left_open_halfline p and
A3: ( for x0 being Real st x0 in left_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in left_open_halfline p holds
diff (f,x0) < 0 ) ; ::_thesis: rng (f | (left_open_halfline p)) is open
A4: f | (left_open_halfline p) is continuous by A2, FDIFF_1:25;
now__::_thesis:_rng_(f_|_(left_open_halfline_p))_is_open
percases ( for x0 being Real st x0 in left_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in left_open_halfline p holds
diff (f,x0) < 0 ) by A3;
suppose for x0 being Real st x0 in left_open_halfline p holds
0 < diff (f,x0) ; ::_thesis: rng (f | (left_open_halfline p)) is open
then f | (left_open_halfline p) is increasing by A1, A2, Th29;
hence rng (f | (left_open_halfline p)) is open by A1, A4, FCONT_3:24; ::_thesis: verum
end;
suppose for x0 being Real st x0 in left_open_halfline p holds
diff (f,x0) < 0 ; ::_thesis: rng (f | (left_open_halfline p)) is open
then f | (left_open_halfline p) is decreasing by A1, A2, Th30;
hence rng (f | (left_open_halfline p)) is open by A1, A4, FCONT_3:24; ::_thesis: verum
end;
end;
end;
hence rng (f | (left_open_halfline p)) is open ; ::_thesis: verum
end;
theorem Th43: :: FDIFF_2:43
for p being Real
for f being PartFunc of REAL,REAL st right_open_halfline p c= dom f & f is_differentiable_on right_open_halfline p & ( for x0 being Real st x0 in right_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in right_open_halfline p holds
diff (f,x0) < 0 ) holds
rng (f | (right_open_halfline p)) is open
proof
let p be Real; ::_thesis: for f being PartFunc of REAL,REAL st right_open_halfline p c= dom f & f is_differentiable_on right_open_halfline p & ( for x0 being Real st x0 in right_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in right_open_halfline p holds
diff (f,x0) < 0 ) holds
rng (f | (right_open_halfline p)) is open
let f be PartFunc of REAL,REAL; ::_thesis: ( right_open_halfline p c= dom f & f is_differentiable_on right_open_halfline p & ( for x0 being Real st x0 in right_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in right_open_halfline p holds
diff (f,x0) < 0 ) implies rng (f | (right_open_halfline p)) is open )
set l = right_open_halfline p;
assume A1: right_open_halfline p c= dom f ; ::_thesis: ( not f is_differentiable_on right_open_halfline p or ( ex x0 being Real st
( x0 in right_open_halfline p & not 0 < diff (f,x0) ) & ex x0 being Real st
( x0 in right_open_halfline p & not diff (f,x0) < 0 ) ) or rng (f | (right_open_halfline p)) is open )
assume that
A2: f is_differentiable_on right_open_halfline p and
A3: ( for x0 being Real st x0 in right_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in right_open_halfline p holds
diff (f,x0) < 0 ) ; ::_thesis: rng (f | (right_open_halfline p)) is open
A4: f | (right_open_halfline p) is continuous by A2, FDIFF_1:25;
now__::_thesis:_rng_(f_|_(right_open_halfline_p))_is_open
percases ( for x0 being Real st x0 in right_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in right_open_halfline p holds
diff (f,x0) < 0 ) by A3;
suppose for x0 being Real st x0 in right_open_halfline p holds
0 < diff (f,x0) ; ::_thesis: rng (f | (right_open_halfline p)) is open
then f | (right_open_halfline p) is increasing by A1, A2, Th33;
hence rng (f | (right_open_halfline p)) is open by A1, A4, FCONT_3:25; ::_thesis: verum
end;
suppose for x0 being Real st x0 in right_open_halfline p holds
diff (f,x0) < 0 ; ::_thesis: rng (f | (right_open_halfline p)) is open
then f | (right_open_halfline p) is decreasing by A1, A2, Th34;
hence rng (f | (right_open_halfline p)) is open by A1, A4, FCONT_3:25; ::_thesis: verum
end;
end;
end;
hence rng (f | (right_open_halfline p)) is open ; ::_thesis: verum
end;
theorem Th44: :: FDIFF_2:44
for f being PartFunc of REAL,REAL st [#] REAL c= dom f & f is_differentiable_on [#] REAL & ( for x0 being Real holds 0 < diff (f,x0) or for x0 being Real holds diff (f,x0) < 0 ) holds
rng f is open
proof
let f be PartFunc of REAL,REAL; ::_thesis: ( [#] REAL c= dom f & f is_differentiable_on [#] REAL & ( for x0 being Real holds 0 < diff (f,x0) or for x0 being Real holds diff (f,x0) < 0 ) implies rng f is open )
assume A1: [#] REAL c= dom f ; ::_thesis: ( not f is_differentiable_on [#] REAL or ( not for x0 being Real holds 0 < diff (f,x0) & not for x0 being Real holds diff (f,x0) < 0 ) or rng f is open )
assume that
A2: f is_differentiable_on [#] REAL and
A3: ( for x0 being Real holds 0 < diff (f,x0) or for x0 being Real holds diff (f,x0) < 0 ) ; ::_thesis: rng f is open
A4: f | ([#] REAL) is continuous by A2, FDIFF_1:25;
now__::_thesis:_rng_f_is_open
percases ( for x0 being Real holds 0 < diff (f,x0) or for x0 being Real holds diff (f,x0) < 0 ) by A3;
suppose for x0 being Real holds 0 < diff (f,x0) ; ::_thesis: rng f is open
then f | ([#] REAL) is increasing by A1, A2, Th37;
hence rng f is open by A1, A4, FCONT_3:26; ::_thesis: verum
end;
suppose for x0 being Real holds diff (f,x0) < 0 ; ::_thesis: rng f is open
then f | ([#] REAL) is decreasing by A1, A2, Th38;
hence rng f is open by A1, A4, FCONT_3:26; ::_thesis: verum
end;
end;
end;
hence rng f is open ; ::_thesis: verum
end;
theorem :: FDIFF_2:45
for f being one-to-one PartFunc of REAL,REAL st [#] REAL c= dom f & f is_differentiable_on [#] REAL & ( for x0 being Real holds 0 < diff (f,x0) or for x0 being Real holds diff (f,x0) < 0 ) holds
( f is one-to-one & f " is_differentiable_on dom (f ") & ( for x0 being Real st x0 in dom (f ") holds
diff ((f "),x0) = 1 / (diff (f,((f ") . x0))) ) )
proof
let f be one-to-one PartFunc of REAL,REAL; ::_thesis: ( [#] REAL c= dom f & f is_differentiable_on [#] REAL & ( for x0 being Real holds 0 < diff (f,x0) or for x0 being Real holds diff (f,x0) < 0 ) implies ( f is one-to-one & f " is_differentiable_on dom (f ") & ( for x0 being Real st x0 in dom (f ") holds
diff ((f "),x0) = 1 / (diff (f,((f ") . x0))) ) ) )
assume that
A1: [#] REAL c= dom f and
A2: f is_differentiable_on [#] REAL and
A3: ( for x0 being Real holds 0 < diff (f,x0) or for x0 being Real holds diff (f,x0) < 0 ) ; ::_thesis: ( f is one-to-one & f " is_differentiable_on dom (f ") & ( for x0 being Real st x0 in dom (f ") holds
diff ((f "),x0) = 1 / (diff (f,((f ") . x0))) ) )
A4: rng f is open by A1, A2, A3, Th44;
thus f is one-to-one ; ::_thesis: ( f " is_differentiable_on dom (f ") & ( for x0 being Real st x0 in dom (f ") holds
diff ((f "),x0) = 1 / (diff (f,((f ") . x0))) ) )
A5: dom (f ") = rng f by FUNCT_1:33;
A6: rng (f ") = dom f by FUNCT_1:33;
A7: for y0 being Real st y0 in dom (f ") holds
( f " is_differentiable_in y0 & diff ((f "),y0) = 1 / (diff (f,((f ") . y0))) )
proof
let y0 be Real; ::_thesis: ( y0 in dom (f ") implies ( f " is_differentiable_in y0 & diff ((f "),y0) = 1 / (diff (f,((f ") . y0))) ) )
assume A8: y0 in dom (f ") ; ::_thesis: ( f " is_differentiable_in y0 & diff ((f "),y0) = 1 / (diff (f,((f ") . y0))) )
then consider x0 being Real such that
A9: x0 in dom f and
A10: y0 = f . x0 by A5, PARTFUN1:3;
A11: for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {y0} & rng (h + c) c= dom (f ") holds
( (h ") (#) (((f ") /* (h + c)) - ((f ") /* c)) is convergent & lim ((h ") (#) (((f ") /* (h + c)) - ((f ") /* c))) = 1 / (diff (f,((f ") . y0))) )
proof
reconsider a = NAT --> ((f ") . y0) as Real_Sequence by FUNCOP_1:45;
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: for c being V8() Real_Sequence st rng c = {y0} & rng (h + c) c= dom (f ") holds
( (h ") (#) (((f ") /* (h + c)) - ((f ") /* c)) is convergent & lim ((h ") (#) (((f ") /* (h + c)) - ((f ") /* c))) = 1 / (diff (f,((f ") . y0))) )
let c be V8() Real_Sequence; ::_thesis: ( rng c = {y0} & rng (h + c) c= dom (f ") implies ( (h ") (#) (((f ") /* (h + c)) - ((f ") /* c)) is convergent & lim ((h ") (#) (((f ") /* (h + c)) - ((f ") /* c))) = 1 / (diff (f,((f ") . y0))) ) )
assume that
A12: rng c = {y0} and
A13: rng (h + c) c= dom (f ") ; ::_thesis: ( (h ") (#) (((f ") /* (h + c)) - ((f ") /* c)) is convergent & lim ((h ") (#) (((f ") /* (h + c)) - ((f ") /* c))) = 1 / (diff (f,((f ") . y0))) )
A14: lim (h + c) = y0 by A12, Th4;
reconsider a = a as V8() Real_Sequence ;
defpred S1[ Element of NAT , real number ] means for r1, r2 being real number st r1 = (h + c) . $1 & r2 = a . $1 holds
r1 = f . (r2 + $2);
A15: for n being Element of NAT ex r being Real st S1[n,r]
proof
let n be Element of NAT ; ::_thesis: ex r being Real st S1[n,r]
(h + c) . n in rng (h + c) by VALUED_0:28;
then consider g being Real such that
g in dom f and
A16: (h + c) . n = f . g by A5, A13, PARTFUN1:3;
take r = g - x0; ::_thesis: S1[n,r]
let r1, r2 be real number ; ::_thesis: ( r1 = (h + c) . n & r2 = a . n implies r1 = f . (r2 + r) )
assume that
A17: r1 = (h + c) . n and
A18: r2 = a . n ; ::_thesis: r1 = f . (r2 + r)
a . n = (f ") . (f . x0) by A10, FUNCOP_1:7
.= x0 by A9, FUNCT_1:34 ;
hence r1 = f . (r2 + r) by A16, A17, A18; ::_thesis: verum
end;
consider b being Real_Sequence such that
A19: for n being Element of NAT holds S1[n,b . n] from FUNCT_2:sch_3(A15);
A20: now__::_thesis:_for_n_being_Element_of_NAT_holds_c_._n_=_f_._x0
let n be Element of NAT ; ::_thesis: c . n = f . x0
c . n in rng c by VALUED_0:28;
hence c . n = f . x0 by A10, A12, TARSKI:def_1; ::_thesis: verum
end;
now__::_thesis:_for_n_being_Element_of_NAT_holds_not_b_._n_=_0
given n being Element of NAT such that A22: b . n = 0 ; ::_thesis: contradiction
A23: (h + c) . n = (h . n) + (c . n) by SEQ_1:7
.= (h . n) + (f . x0) by A20 ;
f . ((a . n) + (b . n)) = f . ((f ") . (f . x0)) by A10, A22, FUNCOP_1:7
.= f . x0 by A9, FUNCT_1:34 ;
then (h . n) + (f . x0) = f . x0 by A19, A23;
hence contradiction by SEQ_1:5; ::_thesis: verum
end;
then A24: b is non-zero by SEQ_1:5;
A25: [#] REAL c= dom f by A2, FDIFF_1:def_6;
then dom f = REAL by XBOOLE_0:def_10;
then A26: f is total by PARTFUN1:def_2;
A27: y0 in dom ((f ") | (rng f)) by A5, A8, RELAT_1:69;
( f | ([#] REAL) is increasing or f | ([#] REAL) is decreasing ) by A1, A2, A3, Th37, Th38;
then (f ") | (rng f) is continuous by A26, FCONT_3:22;
then (f ") | (dom (f ")) is_continuous_in y0 by A5, A27, FCONT_1:def_2;
then A28: f " is_continuous_in y0 by RELAT_1:68;
A29: now__::_thesis:_for_n_being_Element_of_NAT_holds_(((f_")_/*_(h_+_c))_-_a)_._n_=_b_._n
let n be Element of NAT ; ::_thesis: (((f ") /* (h + c)) - a) . n = b . n
A30: (b . n) + (a . n) in [#] REAL ;
thus (((f ") /* (h + c)) - a) . n = (((f ") /* (h + c)) . n) - (a . n) by RFUNCT_2:1
.= ((f ") . ((h + c) . n)) - (a . n) by A13, FUNCT_2:108
.= ((f ") . (f . ((b . n) + (a . n)))) - (a . n) by A19
.= ((b . n) + (a . n)) - (a . n) by A25, A30, FUNCT_1:34
.= b . n ; ::_thesis: verum
end;
A31: h + c is convergent by A12, Th4;
then A32: (f ") /* (h + c) is convergent by A13, A14, A28, FCONT_1:def_1;
then ((f ") /* (h + c)) - a is convergent by SEQ_2:11;
then A33: b is convergent by A29, FUNCT_2:63;
A34: lim a = a . 0 by SEQ_4:26
.= (f ") . y0 by FUNCOP_1:7 ;
(f ") . y0 = lim ((f ") /* (h + c)) by A13, A31, A14, A28, FCONT_1:def_1;
then lim (((f ") /* (h + c)) - a) = ((f ") . y0) - ((f ") . y0) by A32, A34, SEQ_2:12
.= 0 ;
then A35: lim b = 0 by A29, FUNCT_2:63;
A36: rng (b + a) c= dom f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (b + a) or x in dom f )
assume x in rng (b + a) ; ::_thesis: x in dom f
then x in REAL ;
hence x in dom f by A25; ::_thesis: verum
end;
reconsider b = b as non-zero 0 -convergent Real_Sequence by A24, A33, A35, FDIFF_1:def_1;
A37: b " is non-zero by SEQ_1:33;
A38: rng a = {((f ") . y0)}
proof
thus rng a c= {((f ") . y0)} :: according to XBOOLE_0:def_10 ::_thesis: {((f ") . y0)} c= rng a
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng a or x in {((f ") . y0)} )
assume x in rng a ; ::_thesis: x in {((f ") . y0)}
then ex n being Element of NAT st x = a . n by FUNCT_2:113;
then x = (f ") . y0 by FUNCOP_1:7;
hence x in {((f ") . y0)} by TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((f ") . y0)} or x in rng a )
assume x in {((f ") . y0)} ; ::_thesis: x in rng a
then x = (f ") . y0 by TARSKI:def_1;
then a . 0 = x by FUNCOP_1:7;
hence x in rng a by VALUED_0:28; ::_thesis: verum
end;
A39: rng a c= dom f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng a or x in dom f )
assume x in rng a ; ::_thesis: x in dom f
then x = (f ") . y0 by A38, TARSKI:def_1;
hence x in dom f by A6, A8, FUNCT_1:def_3; ::_thesis: verum
end;
now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_a)_._n_=_c_._n
let n be Element of NAT ; ::_thesis: (f /* a) . n = c . n
c . n in rng c by VALUED_0:28;
then A40: c . n = y0 by A12, TARSKI:def_1;
thus (f /* a) . n = f . (a . n) by A39, FUNCT_2:108
.= f . ((f ") . y0) by FUNCOP_1:7
.= c . n by A5, A8, A40, FUNCT_1:35 ; ::_thesis: verum
end;
then A41: f /* a = c by FUNCT_2:63;
now__::_thesis:_for_n_being_Element_of_NAT_holds_h_._n_=_((f_/*_(b_+_a))_-_(f_/*_a))_._n
let n be Element of NAT ; ::_thesis: h . n = ((f /* (b + a)) - (f /* a)) . n
(h + c) . n = f . ((a . n) + (b . n)) by A19;
then (h . n) + (c . n) = f . ((a . n) + (b . n)) by SEQ_1:7;
hence h . n = (f . ((b . n) + (a . n))) - ((f /* a) . n) by A41
.= (f . ((b + a) . n)) - ((f /* a) . n) by SEQ_1:7
.= ((f /* (b + a)) . n) - ((f /* a) . n) by A36, FUNCT_2:108
.= ((f /* (b + a)) - (f /* a)) . n by RFUNCT_2:1 ;
::_thesis: verum
end;
then A42: h = (f /* (b + a)) - (f /* a) by FUNCT_2:63;
then (f /* (b + a)) - (f /* a) is non-zero ;
then A43: (b ") (#) ((f /* (b + a)) - (f /* a)) is non-zero by A37, SEQ_1:35;
A44: rng c c= dom (f ")
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng c or x in dom (f ") )
assume x in rng c ; ::_thesis: x in dom (f ")
hence x in dom (f ") by A8, A12, TARSKI:def_1; ::_thesis: verum
end;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((h_")_(#)_(((f_")_/*_(h_+_c))_-_((f_")_/*_c)))_._n_=_(((b_")_(#)_((f_/*_(b_+_a))_-_(f_/*_a)))_")_._n
let n be Element of NAT ; ::_thesis: ((h ") (#) (((f ") /* (h + c)) - ((f ") /* c))) . n = (((b ") (#) ((f /* (b + a)) - (f /* a))) ") . n
A45: (a . n) + (b . n) in [#] REAL ;
c . n in rng c by VALUED_0:28;
then A46: c . n = y0 by A12, TARSKI:def_1;
thus ((h ") (#) (((f ") /* (h + c)) - ((f ") /* c))) . n = ((h ") . n) * ((((f ") /* (h + c)) - ((f ") /* c)) . n) by SEQ_1:8
.= ((h ") . n) * ((((f ") /* (h + c)) . n) - (((f ") /* c) . n)) by RFUNCT_2:1
.= ((h ") . n) * (((f ") . ((h + c) . n)) - (((f ") /* c) . n)) by A13, FUNCT_2:108
.= ((h ") . n) * (((f ") . (f . ((a . n) + (b . n)))) - (((f ") /* c) . n)) by A19
.= ((h ") . n) * (((a . n) + (b . n)) - (((f ") /* c) . n)) by A25, A45, FUNCT_1:34
.= ((h ") . n) * (((a . n) + (b . n)) - ((f ") . (c . n))) by A44, FUNCT_2:108
.= ((h ") . n) * (((a . n) + (b . n)) - (a . n)) by A46, FUNCOP_1:7
.= ((h ") (#) ((b ") ")) . n by SEQ_1:8
.= (((b ") (#) ((f /* (b + a)) - (f /* a))) ") . n by A42, SEQ_1:36 ; ::_thesis: verum
end;
then A47: (h ") (#) (((f ") /* (h + c)) - ((f ") /* c)) = ((b ") (#) ((f /* (b + a)) - (f /* a))) " by FUNCT_2:63;
A48: f is_differentiable_in (f ") . y0 by A2, FDIFF_1:9;
then A49: lim ((b ") (#) ((f /* (b + a)) - (f /* a))) = diff (f,((f ") . y0)) by A38, A36, Th12;
diff (f,((f ") . y0)) = diff (f,((f ") . y0)) ;
then A50: (b ") (#) ((f /* (b + a)) - (f /* a)) is convergent by A38, A36, A48, Th12;
A51: 0 <> diff (f,((f ") . y0)) by A3;
hence (h ") (#) (((f ") /* (h + c)) - ((f ") /* c)) is convergent by A47, A43, A50, A49, SEQ_2:21; ::_thesis: lim ((h ") (#) (((f ") /* (h + c)) - ((f ") /* c))) = 1 / (diff (f,((f ") . y0)))
thus lim ((h ") (#) (((f ") /* (h + c)) - ((f ") /* c))) = (diff (f,((f ") . y0))) " by A47, A43, A50, A49, A51, SEQ_2:22
.= 1 / (diff (f,((f ") . y0))) by XCMPLX_1:215 ; ::_thesis: verum
end;
ex N being Neighbourhood of y0 st N c= dom (f ") by A4, A5, A8, RCOMP_1:18;
hence ( f " is_differentiable_in y0 & diff ((f "),y0) = 1 / (diff (f,((f ") . y0))) ) by A11, Th12; ::_thesis: verum
end;
then for y0 being Real st y0 in dom (f ") holds
f " is_differentiable_in y0 ;
hence f " is_differentiable_on dom (f ") by A4, A5, FDIFF_1:9; ::_thesis: for x0 being Real st x0 in dom (f ") holds
diff ((f "),x0) = 1 / (diff (f,((f ") . x0)))
let x0 be Real; ::_thesis: ( x0 in dom (f ") implies diff ((f "),x0) = 1 / (diff (f,((f ") . x0))) )
assume x0 in dom (f ") ; ::_thesis: diff ((f "),x0) = 1 / (diff (f,((f ") . x0)))
hence diff ((f "),x0) = 1 / (diff (f,((f ") . x0))) by A7; ::_thesis: verum
end;
theorem :: FDIFF_2:46
for p being Real
for f being one-to-one PartFunc of REAL,REAL st left_open_halfline p c= dom f & f is_differentiable_on left_open_halfline p & ( for x0 being Real st x0 in left_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in left_open_halfline p holds
diff (f,x0) < 0 ) holds
( f | (left_open_halfline p) is one-to-one & (f | (left_open_halfline p)) " is_differentiable_on dom ((f | (left_open_halfline p)) ") & ( for x0 being Real st x0 in dom ((f | (left_open_halfline p)) ") holds
diff (((f | (left_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . x0))) ) )
proof
let p be Real; ::_thesis: for f being one-to-one PartFunc of REAL,REAL st left_open_halfline p c= dom f & f is_differentiable_on left_open_halfline p & ( for x0 being Real st x0 in left_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in left_open_halfline p holds
diff (f,x0) < 0 ) holds
( f | (left_open_halfline p) is one-to-one & (f | (left_open_halfline p)) " is_differentiable_on dom ((f | (left_open_halfline p)) ") & ( for x0 being Real st x0 in dom ((f | (left_open_halfline p)) ") holds
diff (((f | (left_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . x0))) ) )
let f be one-to-one PartFunc of REAL,REAL; ::_thesis: ( left_open_halfline p c= dom f & f is_differentiable_on left_open_halfline p & ( for x0 being Real st x0 in left_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in left_open_halfline p holds
diff (f,x0) < 0 ) implies ( f | (left_open_halfline p) is one-to-one & (f | (left_open_halfline p)) " is_differentiable_on dom ((f | (left_open_halfline p)) ") & ( for x0 being Real st x0 in dom ((f | (left_open_halfline p)) ") holds
diff (((f | (left_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . x0))) ) ) )
set l = left_open_halfline p;
assume that
A1: left_open_halfline p c= dom f and
A2: f is_differentiable_on left_open_halfline p and
A3: ( for x0 being Real st x0 in left_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in left_open_halfline p holds
diff (f,x0) < 0 ) ; ::_thesis: ( f | (left_open_halfline p) is one-to-one & (f | (left_open_halfline p)) " is_differentiable_on dom ((f | (left_open_halfline p)) ") & ( for x0 being Real st x0 in dom ((f | (left_open_halfline p)) ") holds
diff (((f | (left_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . x0))) ) )
A4: rng (f | (left_open_halfline p)) is open by A1, A2, A3, Th42;
set f1 = f | (left_open_halfline p);
thus f | (left_open_halfline p) is one-to-one ; ::_thesis: ( (f | (left_open_halfline p)) " is_differentiable_on dom ((f | (left_open_halfline p)) ") & ( for x0 being Real st x0 in dom ((f | (left_open_halfline p)) ") holds
diff (((f | (left_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . x0))) ) )
A5: dom ((f | (left_open_halfline p)) ") = rng (f | (left_open_halfline p)) by FUNCT_1:33;
A6: rng ((f | (left_open_halfline p)) ") = dom (f | (left_open_halfline p)) by FUNCT_1:33;
A7: for y0 being Real st y0 in dom ((f | (left_open_halfline p)) ") holds
( (f | (left_open_halfline p)) " is_differentiable_in y0 & diff (((f | (left_open_halfline p)) "),y0) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . y0))) )
proof
let y0 be Real; ::_thesis: ( y0 in dom ((f | (left_open_halfline p)) ") implies ( (f | (left_open_halfline p)) " is_differentiable_in y0 & diff (((f | (left_open_halfline p)) "),y0) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . y0))) ) )
assume A8: y0 in dom ((f | (left_open_halfline p)) ") ; ::_thesis: ( (f | (left_open_halfline p)) " is_differentiable_in y0 & diff (((f | (left_open_halfline p)) "),y0) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . y0))) )
then consider x0 being Real such that
A9: x0 in dom (f | (left_open_halfline p)) and
A10: y0 = (f | (left_open_halfline p)) . x0 by A5, PARTFUN1:3;
A11: for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {y0} & rng (h + c) c= dom ((f | (left_open_halfline p)) ") holds
( (h ") (#) ((((f | (left_open_halfline p)) ") /* (h + c)) - (((f | (left_open_halfline p)) ") /* c)) is convergent & lim ((h ") (#) ((((f | (left_open_halfline p)) ") /* (h + c)) - (((f | (left_open_halfline p)) ") /* c))) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . y0))) )
proof
A12: left_open_halfline p c= dom f by A2, FDIFF_1:def_6;
( f | (left_open_halfline p) is increasing or f | (left_open_halfline p) is decreasing ) by A1, A2, A3, Th29, Th30;
then ((f | (left_open_halfline p)) ") | (f .: (left_open_halfline p)) is continuous by A12, FCONT_3:18;
then A13: ((f | (left_open_halfline p)) ") | (rng (f | (left_open_halfline p))) is continuous by RELAT_1:115;
y0 in dom (((f | (left_open_halfline p)) ") | (rng (f | (left_open_halfline p)))) by A5, A8, RELAT_1:69;
then ((f | (left_open_halfline p)) ") | (rng (f | (left_open_halfline p))) is_continuous_in y0 by A13, FCONT_1:def_2;
then A14: (f | (left_open_halfline p)) " is_continuous_in y0 by A5, RELAT_1:68;
reconsider a = NAT --> (((f | (left_open_halfline p)) ") . y0) as Real_Sequence by FUNCOP_1:45;
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: for c being V8() Real_Sequence st rng c = {y0} & rng (h + c) c= dom ((f | (left_open_halfline p)) ") holds
( (h ") (#) ((((f | (left_open_halfline p)) ") /* (h + c)) - (((f | (left_open_halfline p)) ") /* c)) is convergent & lim ((h ") (#) ((((f | (left_open_halfline p)) ") /* (h + c)) - (((f | (left_open_halfline p)) ") /* c))) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . y0))) )
let c be V8() Real_Sequence; ::_thesis: ( rng c = {y0} & rng (h + c) c= dom ((f | (left_open_halfline p)) ") implies ( (h ") (#) ((((f | (left_open_halfline p)) ") /* (h + c)) - (((f | (left_open_halfline p)) ") /* c)) is convergent & lim ((h ") (#) ((((f | (left_open_halfline p)) ") /* (h + c)) - (((f | (left_open_halfline p)) ") /* c))) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . y0))) ) )
assume that
A15: rng c = {y0} and
A16: rng (h + c) c= dom ((f | (left_open_halfline p)) ") ; ::_thesis: ( (h ") (#) ((((f | (left_open_halfline p)) ") /* (h + c)) - (((f | (left_open_halfline p)) ") /* c)) is convergent & lim ((h ") (#) ((((f | (left_open_halfline p)) ") /* (h + c)) - (((f | (left_open_halfline p)) ") /* c))) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . y0))) )
A17: lim (h + c) = y0 by A15, Th4;
reconsider a = a as V8() Real_Sequence ;
defpred S1[ Element of NAT , real number ] means for r1, r2 being real number st r1 = (h + c) . $1 & r2 = a . $1 holds
( r1 = f . (r2 + $2) & r2 + $2 in dom f & r2 + $2 in dom (f | (left_open_halfline p)) );
A18: for n being Element of NAT ex r being Real st S1[n,r]
proof
let n be Element of NAT ; ::_thesis: ex r being Real st S1[n,r]
(h + c) . n in rng (h + c) by VALUED_0:28;
then consider g being Real such that
A19: g in dom (f | (left_open_halfline p)) and
A20: (h + c) . n = (f | (left_open_halfline p)) . g by A5, A16, PARTFUN1:3;
take r = g - x0; ::_thesis: S1[n,r]
let r1, r2 be real number ; ::_thesis: ( r1 = (h + c) . n & r2 = a . n implies ( r1 = f . (r2 + r) & r2 + r in dom f & r2 + r in dom (f | (left_open_halfline p)) ) )
assume that
A21: r1 = (h + c) . n and
A22: r2 = a . n ; ::_thesis: ( r1 = f . (r2 + r) & r2 + r in dom f & r2 + r in dom (f | (left_open_halfline p)) )
A23: a . n = ((f | (left_open_halfline p)) ") . ((f | (left_open_halfline p)) . x0) by A10, FUNCOP_1:7
.= x0 by A9, FUNCT_1:34 ;
hence r1 = f . (r2 + r) by A19, A20, A21, A22, FUNCT_1:47; ::_thesis: ( r2 + r in dom f & r2 + r in dom (f | (left_open_halfline p)) )
g in (dom f) /\ (left_open_halfline p) by A19, RELAT_1:61;
hence ( r2 + r in dom f & r2 + r in dom (f | (left_open_halfline p)) ) by A19, A23, A22, XBOOLE_0:def_4; ::_thesis: verum
end;
consider b being Real_Sequence such that
A24: for n being Element of NAT holds S1[n,b . n] from FUNCT_2:sch_3(A18);
A25: now__::_thesis:_for_n_being_Element_of_NAT_holds_((((f_|_(left_open_halfline_p))_")_/*_(h_+_c))_-_a)_._n_=_b_._n
let n be Element of NAT ; ::_thesis: ((((f | (left_open_halfline p)) ") /* (h + c)) - a) . n = b . n
A26: (h + c) . n = (h + c) . n ;
then A27: (a . n) + (b . n) in dom (f | (left_open_halfline p)) by A24;
thus ((((f | (left_open_halfline p)) ") /* (h + c)) - a) . n = ((((f | (left_open_halfline p)) ") /* (h + c)) . n) - (a . n) by RFUNCT_2:1
.= (((f | (left_open_halfline p)) ") . ((h + c) . n)) - (a . n) by A16, FUNCT_2:108
.= (((f | (left_open_halfline p)) ") . (f . ((a . n) + (b . n)))) - (a . n) by A24
.= (((f | (left_open_halfline p)) ") . ((f | (left_open_halfline p)) . ((a . n) + (b . n)))) - (a . n) by A24, A26, FUNCT_1:47
.= ((a . n) + (b . n)) - (a . n) by A27, FUNCT_1:34
.= b . n ; ::_thesis: verum
end;
A28: h + c is convergent by A15, Th4;
then A29: ((f | (left_open_halfline p)) ") /* (h + c) is convergent by A16, A17, A14, FCONT_1:def_1;
then (((f | (left_open_halfline p)) ") /* (h + c)) - a is convergent by SEQ_2:11;
then A30: b is convergent by A25, FUNCT_2:63;
A31: lim a = a . 0 by SEQ_4:26
.= ((f | (left_open_halfline p)) ") . y0 by FUNCOP_1:7 ;
((f | (left_open_halfline p)) ") . y0 = lim (((f | (left_open_halfline p)) ") /* (h + c)) by A16, A28, A17, A14, FCONT_1:def_1;
then lim ((((f | (left_open_halfline p)) ") /* (h + c)) - a) = (((f | (left_open_halfline p)) ") . y0) - (((f | (left_open_halfline p)) ") . y0) by A29, A31, SEQ_2:12
.= 0 ;
then A32: lim b = 0 by A25, FUNCT_2:63;
A33: rng (b + a) c= dom f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (b + a) or x in dom f )
assume x in rng (b + a) ; ::_thesis: x in dom f
then consider n being Element of NAT such that
A34: x = (b + a) . n by FUNCT_2:113;
A35: (h + c) . n = (h + c) . n ;
x = (a . n) + (b . n) by A34, SEQ_1:7;
hence x in dom f by A24, A35; ::_thesis: verum
end;
((f | (left_open_halfline p)) ") . y0 in dom (f | (left_open_halfline p)) by A6, A8, FUNCT_1:def_3;
then ((f | (left_open_halfline p)) ") . y0 in (dom f) /\ (left_open_halfline p) by RELAT_1:61;
then A36: ((f | (left_open_halfline p)) ") . y0 in left_open_halfline p by XBOOLE_0:def_4;
then A37: f is_differentiable_in ((f | (left_open_halfline p)) ") . y0 by A2, FDIFF_1:9;
A38: now__::_thesis:_for_n_being_Element_of_NAT_holds_c_._n_=_(f_|_(left_open_halfline_p))_._x0
let n be Element of NAT ; ::_thesis: c . n = (f | (left_open_halfline p)) . x0
c . n in rng c by VALUED_0:28;
hence c . n = (f | (left_open_halfline p)) . x0 by A10, A15, TARSKI:def_1; ::_thesis: verum
end;
A39: 0 <> diff (f,(((f | (left_open_halfline p)) ") . y0)) by A3, A36;
now__::_thesis:_for_n_being_Element_of_NAT_holds_not_b_._n_=_0
given n being Element of NAT such that A41: b . n = 0 ; ::_thesis: contradiction
a . n = ((f | (left_open_halfline p)) ") . ((f | (left_open_halfline p)) . x0) by A10, FUNCOP_1:7
.= x0 by A9, FUNCT_1:34 ;
then A42: f . ((a . n) + (b . n)) = (f | (left_open_halfline p)) . x0 by A9, A41, FUNCT_1:47;
(h + c) . n = (h . n) + (c . n) by SEQ_1:7
.= (h . n) + ((f | (left_open_halfline p)) . x0) by A38 ;
then (h . n) + ((f | (left_open_halfline p)) . x0) = (f | (left_open_halfline p)) . x0 by A24, A42;
hence contradiction by SEQ_1:5; ::_thesis: verum
end;
then b is non-zero by SEQ_1:5;
then reconsider b = b as non-zero 0 -convergent Real_Sequence by A30, A32, FDIFF_1:def_1;
A44: b " is non-zero by SEQ_1:33;
A45: rng a = {(((f | (left_open_halfline p)) ") . y0)}
proof
thus rng a c= {(((f | (left_open_halfline p)) ") . y0)} :: according to XBOOLE_0:def_10 ::_thesis: {(((f | (left_open_halfline p)) ") . y0)} c= rng a
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng a or x in {(((f | (left_open_halfline p)) ") . y0)} )
assume x in rng a ; ::_thesis: x in {(((f | (left_open_halfline p)) ") . y0)}
then ex n being Element of NAT st x = a . n by FUNCT_2:113;
then x = ((f | (left_open_halfline p)) ") . y0 by FUNCOP_1:7;
hence x in {(((f | (left_open_halfline p)) ") . y0)} by TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(((f | (left_open_halfline p)) ") . y0)} or x in rng a )
assume x in {(((f | (left_open_halfline p)) ") . y0)} ; ::_thesis: x in rng a
then x = ((f | (left_open_halfline p)) ") . y0 by TARSKI:def_1;
then a . 0 = x by FUNCOP_1:7;
hence x in rng a by VALUED_0:28; ::_thesis: verum
end;
A46: rng a c= dom f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng a or x in dom f )
assume x in rng a ; ::_thesis: x in dom f
then x = ((f | (left_open_halfline p)) ") . y0 by A45, TARSKI:def_1;
then x = x0 by A9, A10, FUNCT_1:34;
then x in (dom f) /\ (left_open_halfline p) by A9, RELAT_1:61;
hence x in dom f by XBOOLE_0:def_4; ::_thesis: verum
end;
now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_a)_._n_=_c_._n
let n be Element of NAT ; ::_thesis: (f /* a) . n = c . n
A47: ((f | (left_open_halfline p)) ") . y0 in rng ((f | (left_open_halfline p)) ") by A8, FUNCT_1:def_3;
c . n in rng c by VALUED_0:28;
then A48: c . n = y0 by A15, TARSKI:def_1;
thus (f /* a) . n = f . (a . n) by A46, FUNCT_2:108
.= f . (((f | (left_open_halfline p)) ") . y0) by FUNCOP_1:7
.= (f | (left_open_halfline p)) . (((f | (left_open_halfline p)) ") . y0) by A6, A47, FUNCT_1:47
.= c . n by A5, A8, A48, FUNCT_1:35 ; ::_thesis: verum
end;
then A49: f /* a = c by FUNCT_2:63;
now__::_thesis:_for_n_being_Element_of_NAT_holds_h_._n_=_((f_/*_(b_+_a))_-_(f_/*_a))_._n
let n be Element of NAT ; ::_thesis: h . n = ((f /* (b + a)) - (f /* a)) . n
(h + c) . n = f . ((a . n) + (b . n)) by A24;
then (h . n) + (c . n) = f . ((a . n) + (b . n)) by SEQ_1:7;
hence h . n = (f . ((b . n) + (a . n))) - ((f /* a) . n) by A49
.= (f . ((b + a) . n)) - ((f /* a) . n) by SEQ_1:7
.= ((f /* (b + a)) . n) - ((f /* a) . n) by A33, FUNCT_2:108
.= ((f /* (b + a)) - (f /* a)) . n by RFUNCT_2:1 ;
::_thesis: verum
end;
then A50: h = (f /* (b + a)) - (f /* a) by FUNCT_2:63;
then (f /* (b + a)) - (f /* a) is non-zero ;
then A51: (b ") (#) ((f /* (b + a)) - (f /* a)) is non-zero by A44, SEQ_1:35;
A52: rng c c= dom ((f | (left_open_halfline p)) ")
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng c or x in dom ((f | (left_open_halfline p)) ") )
assume x in rng c ; ::_thesis: x in dom ((f | (left_open_halfline p)) ")
hence x in dom ((f | (left_open_halfline p)) ") by A8, A15, TARSKI:def_1; ::_thesis: verum
end;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((h_")_(#)_((((f_|_(left_open_halfline_p))_")_/*_(h_+_c))_-_(((f_|_(left_open_halfline_p))_")_/*_c)))_._n_=_(((b_")_(#)_((f_/*_(b_+_a))_-_(f_/*_a)))_")_._n
let n be Element of NAT ; ::_thesis: ((h ") (#) ((((f | (left_open_halfline p)) ") /* (h + c)) - (((f | (left_open_halfline p)) ") /* c))) . n = (((b ") (#) ((f /* (b + a)) - (f /* a))) ") . n
A53: (h + c) . n = (h + c) . n ;
then A54: (a . n) + (b . n) in dom (f | (left_open_halfline p)) by A24;
c . n in rng c by VALUED_0:28;
then A55: c . n = y0 by A15, TARSKI:def_1;
thus ((h ") (#) ((((f | (left_open_halfline p)) ") /* (h + c)) - (((f | (left_open_halfline p)) ") /* c))) . n = ((h ") . n) * (((((f | (left_open_halfline p)) ") /* (h + c)) - (((f | (left_open_halfline p)) ") /* c)) . n) by SEQ_1:8
.= ((h ") . n) * (((((f | (left_open_halfline p)) ") /* (h + c)) . n) - ((((f | (left_open_halfline p)) ") /* c) . n)) by RFUNCT_2:1
.= ((h ") . n) * ((((f | (left_open_halfline p)) ") . ((h + c) . n)) - ((((f | (left_open_halfline p)) ") /* c) . n)) by A16, FUNCT_2:108
.= ((h ") . n) * ((((f | (left_open_halfline p)) ") . (f . ((a . n) + (b . n)))) - ((((f | (left_open_halfline p)) ") /* c) . n)) by A24
.= ((h ") . n) * ((((f | (left_open_halfline p)) ") . ((f | (left_open_halfline p)) . ((a . n) + (b . n)))) - ((((f | (left_open_halfline p)) ") /* c) . n)) by A24, A53, FUNCT_1:47
.= ((h ") . n) * (((a . n) + (b . n)) - ((((f | (left_open_halfline p)) ") /* c) . n)) by A54, FUNCT_1:34
.= ((h ") . n) * (((a . n) + (b . n)) - (((f | (left_open_halfline p)) ") . (c . n))) by A52, FUNCT_2:108
.= ((h ") . n) * (((a . n) + (b . n)) - (a . n)) by A55, FUNCOP_1:7
.= ((h ") (#) ((b ") ")) . n by SEQ_1:8
.= (((b ") (#) ((f /* (b + a)) - (f /* a))) ") . n by A50, SEQ_1:36 ; ::_thesis: verum
end;
then A56: (h ") (#) ((((f | (left_open_halfline p)) ") /* (h + c)) - (((f | (left_open_halfline p)) ") /* c)) = ((b ") (#) ((f /* (b + a)) - (f /* a))) " by FUNCT_2:63;
diff (f,(((f | (left_open_halfline p)) ") . y0)) = diff (f,(((f | (left_open_halfline p)) ") . y0)) ;
then A57: (b ") (#) ((f /* (b + a)) - (f /* a)) is convergent by A45, A33, A37, Th12;
A58: lim ((b ") (#) ((f /* (b + a)) - (f /* a))) = diff (f,(((f | (left_open_halfline p)) ") . y0)) by A45, A33, A37, Th12;
hence (h ") (#) ((((f | (left_open_halfline p)) ") /* (h + c)) - (((f | (left_open_halfline p)) ") /* c)) is convergent by A56, A51, A57, A39, SEQ_2:21; ::_thesis: lim ((h ") (#) ((((f | (left_open_halfline p)) ") /* (h + c)) - (((f | (left_open_halfline p)) ") /* c))) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . y0)))
thus lim ((h ") (#) ((((f | (left_open_halfline p)) ") /* (h + c)) - (((f | (left_open_halfline p)) ") /* c))) = (diff (f,(((f | (left_open_halfline p)) ") . y0))) " by A56, A51, A57, A58, A39, SEQ_2:22
.= 1 / (diff (f,(((f | (left_open_halfline p)) ") . y0))) by XCMPLX_1:215 ; ::_thesis: verum
end;
ex N being Neighbourhood of y0 st N c= dom ((f | (left_open_halfline p)) ") by A4, A5, A8, RCOMP_1:18;
hence ( (f | (left_open_halfline p)) " is_differentiable_in y0 & diff (((f | (left_open_halfline p)) "),y0) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . y0))) ) by A11, Th12; ::_thesis: verum
end;
then for y0 being Real st y0 in dom ((f | (left_open_halfline p)) ") holds
(f | (left_open_halfline p)) " is_differentiable_in y0 ;
hence (f | (left_open_halfline p)) " is_differentiable_on dom ((f | (left_open_halfline p)) ") by A4, A5, FDIFF_1:9; ::_thesis: for x0 being Real st x0 in dom ((f | (left_open_halfline p)) ") holds
diff (((f | (left_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . x0)))
let x0 be Real; ::_thesis: ( x0 in dom ((f | (left_open_halfline p)) ") implies diff (((f | (left_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . x0))) )
assume x0 in dom ((f | (left_open_halfline p)) ") ; ::_thesis: diff (((f | (left_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . x0)))
hence diff (((f | (left_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (left_open_halfline p)) ") . x0))) by A7; ::_thesis: verum
end;
theorem :: FDIFF_2:47
for p being Real
for f being one-to-one PartFunc of REAL,REAL st right_open_halfline p c= dom f & f is_differentiable_on right_open_halfline p & ( for x0 being Real st x0 in right_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in right_open_halfline p holds
diff (f,x0) < 0 ) holds
( f | (right_open_halfline p) is one-to-one & (f | (right_open_halfline p)) " is_differentiable_on dom ((f | (right_open_halfline p)) ") & ( for x0 being Real st x0 in dom ((f | (right_open_halfline p)) ") holds
diff (((f | (right_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . x0))) ) )
proof
let p be Real; ::_thesis: for f being one-to-one PartFunc of REAL,REAL st right_open_halfline p c= dom f & f is_differentiable_on right_open_halfline p & ( for x0 being Real st x0 in right_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in right_open_halfline p holds
diff (f,x0) < 0 ) holds
( f | (right_open_halfline p) is one-to-one & (f | (right_open_halfline p)) " is_differentiable_on dom ((f | (right_open_halfline p)) ") & ( for x0 being Real st x0 in dom ((f | (right_open_halfline p)) ") holds
diff (((f | (right_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . x0))) ) )
let f be one-to-one PartFunc of REAL,REAL; ::_thesis: ( right_open_halfline p c= dom f & f is_differentiable_on right_open_halfline p & ( for x0 being Real st x0 in right_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in right_open_halfline p holds
diff (f,x0) < 0 ) implies ( f | (right_open_halfline p) is one-to-one & (f | (right_open_halfline p)) " is_differentiable_on dom ((f | (right_open_halfline p)) ") & ( for x0 being Real st x0 in dom ((f | (right_open_halfline p)) ") holds
diff (((f | (right_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . x0))) ) ) )
set l = right_open_halfline p;
assume that
A1: right_open_halfline p c= dom f and
A2: f is_differentiable_on right_open_halfline p and
A3: ( for x0 being Real st x0 in right_open_halfline p holds
0 < diff (f,x0) or for x0 being Real st x0 in right_open_halfline p holds
diff (f,x0) < 0 ) ; ::_thesis: ( f | (right_open_halfline p) is one-to-one & (f | (right_open_halfline p)) " is_differentiable_on dom ((f | (right_open_halfline p)) ") & ( for x0 being Real st x0 in dom ((f | (right_open_halfline p)) ") holds
diff (((f | (right_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . x0))) ) )
A4: rng (f | (right_open_halfline p)) is open by A1, A2, A3, Th43;
set f1 = f | (right_open_halfline p);
thus f | (right_open_halfline p) is one-to-one ; ::_thesis: ( (f | (right_open_halfline p)) " is_differentiable_on dom ((f | (right_open_halfline p)) ") & ( for x0 being Real st x0 in dom ((f | (right_open_halfline p)) ") holds
diff (((f | (right_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . x0))) ) )
A5: dom ((f | (right_open_halfline p)) ") = rng (f | (right_open_halfline p)) by FUNCT_1:33;
A6: rng ((f | (right_open_halfline p)) ") = dom (f | (right_open_halfline p)) by FUNCT_1:33;
A7: for y0 being Real st y0 in dom ((f | (right_open_halfline p)) ") holds
( (f | (right_open_halfline p)) " is_differentiable_in y0 & diff (((f | (right_open_halfline p)) "),y0) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . y0))) )
proof
let y0 be Real; ::_thesis: ( y0 in dom ((f | (right_open_halfline p)) ") implies ( (f | (right_open_halfline p)) " is_differentiable_in y0 & diff (((f | (right_open_halfline p)) "),y0) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . y0))) ) )
assume A8: y0 in dom ((f | (right_open_halfline p)) ") ; ::_thesis: ( (f | (right_open_halfline p)) " is_differentiable_in y0 & diff (((f | (right_open_halfline p)) "),y0) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . y0))) )
then consider x0 being Real such that
A9: x0 in dom (f | (right_open_halfline p)) and
A10: y0 = (f | (right_open_halfline p)) . x0 by A5, PARTFUN1:3;
A11: for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {y0} & rng (h + c) c= dom ((f | (right_open_halfline p)) ") holds
( (h ") (#) ((((f | (right_open_halfline p)) ") /* (h + c)) - (((f | (right_open_halfline p)) ") /* c)) is convergent & lim ((h ") (#) ((((f | (right_open_halfline p)) ") /* (h + c)) - (((f | (right_open_halfline p)) ") /* c))) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . y0))) )
proof
A12: right_open_halfline p c= dom f by A2, FDIFF_1:def_6;
( f | (right_open_halfline p) is increasing or f | (right_open_halfline p) is decreasing ) by A1, A2, A3, Th33, Th34;
then ((f | (right_open_halfline p)) ") | (f .: (right_open_halfline p)) is continuous by A12, FCONT_3:19;
then A13: ((f | (right_open_halfline p)) ") | (rng (f | (right_open_halfline p))) is continuous by RELAT_1:115;
y0 in dom (((f | (right_open_halfline p)) ") | (rng (f | (right_open_halfline p)))) by A5, A8, RELAT_1:69;
then ((f | (right_open_halfline p)) ") | (rng (f | (right_open_halfline p))) is_continuous_in y0 by A13, FCONT_1:def_2;
then A14: (f | (right_open_halfline p)) " is_continuous_in y0 by A5, RELAT_1:68;
reconsider a = NAT --> (((f | (right_open_halfline p)) ") . y0) as Real_Sequence by FUNCOP_1:45;
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: for c being V8() Real_Sequence st rng c = {y0} & rng (h + c) c= dom ((f | (right_open_halfline p)) ") holds
( (h ") (#) ((((f | (right_open_halfline p)) ") /* (h + c)) - (((f | (right_open_halfline p)) ") /* c)) is convergent & lim ((h ") (#) ((((f | (right_open_halfline p)) ") /* (h + c)) - (((f | (right_open_halfline p)) ") /* c))) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . y0))) )
let c be V8() Real_Sequence; ::_thesis: ( rng c = {y0} & rng (h + c) c= dom ((f | (right_open_halfline p)) ") implies ( (h ") (#) ((((f | (right_open_halfline p)) ") /* (h + c)) - (((f | (right_open_halfline p)) ") /* c)) is convergent & lim ((h ") (#) ((((f | (right_open_halfline p)) ") /* (h + c)) - (((f | (right_open_halfline p)) ") /* c))) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . y0))) ) )
assume that
A15: rng c = {y0} and
A16: rng (h + c) c= dom ((f | (right_open_halfline p)) ") ; ::_thesis: ( (h ") (#) ((((f | (right_open_halfline p)) ") /* (h + c)) - (((f | (right_open_halfline p)) ") /* c)) is convergent & lim ((h ") (#) ((((f | (right_open_halfline p)) ") /* (h + c)) - (((f | (right_open_halfline p)) ") /* c))) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . y0))) )
A17: lim (h + c) = y0 by A15, Th4;
reconsider a = a as V8() Real_Sequence ;
defpred S1[ Element of NAT , real number ] means for r1, r2 being real number st r1 = (h + c) . $1 & r2 = a . $1 holds
( r1 = f . (r2 + $2) & r2 + $2 in dom f & r2 + $2 in dom (f | (right_open_halfline p)) );
A18: for n being Element of NAT ex r being Real st S1[n,r]
proof
let n be Element of NAT ; ::_thesis: ex r being Real st S1[n,r]
(h + c) . n in rng (h + c) by VALUED_0:28;
then consider g being Real such that
A19: g in dom (f | (right_open_halfline p)) and
A20: (h + c) . n = (f | (right_open_halfline p)) . g by A5, A16, PARTFUN1:3;
take r = g - x0; ::_thesis: S1[n,r]
let r1, r2 be real number ; ::_thesis: ( r1 = (h + c) . n & r2 = a . n implies ( r1 = f . (r2 + r) & r2 + r in dom f & r2 + r in dom (f | (right_open_halfline p)) ) )
assume that
A21: r1 = (h + c) . n and
A22: r2 = a . n ; ::_thesis: ( r1 = f . (r2 + r) & r2 + r in dom f & r2 + r in dom (f | (right_open_halfline p)) )
A23: a . n = ((f | (right_open_halfline p)) ") . ((f | (right_open_halfline p)) . x0) by A10, FUNCOP_1:7
.= x0 by A9, FUNCT_1:34 ;
hence r1 = f . (r2 + r) by A19, A20, A21, A22, FUNCT_1:47; ::_thesis: ( r2 + r in dom f & r2 + r in dom (f | (right_open_halfline p)) )
g in (dom f) /\ (right_open_halfline p) by A19, RELAT_1:61;
hence ( r2 + r in dom f & r2 + r in dom (f | (right_open_halfline p)) ) by A19, A23, A22, XBOOLE_0:def_4; ::_thesis: verum
end;
consider b being Real_Sequence such that
A24: for n being Element of NAT holds S1[n,b . n] from FUNCT_2:sch_3(A18);
A25: now__::_thesis:_for_n_being_Element_of_NAT_holds_((((f_|_(right_open_halfline_p))_")_/*_(h_+_c))_-_a)_._n_=_b_._n
let n be Element of NAT ; ::_thesis: ((((f | (right_open_halfline p)) ") /* (h + c)) - a) . n = b . n
A26: (h + c) . n = (h + c) . n ;
then A27: (a . n) + (b . n) in dom (f | (right_open_halfline p)) by A24;
thus ((((f | (right_open_halfline p)) ") /* (h + c)) - a) . n = ((((f | (right_open_halfline p)) ") /* (h + c)) . n) - (a . n) by RFUNCT_2:1
.= (((f | (right_open_halfline p)) ") . ((h + c) . n)) - (a . n) by A16, FUNCT_2:108
.= (((f | (right_open_halfline p)) ") . (f . ((a . n) + (b . n)))) - (a . n) by A24
.= (((f | (right_open_halfline p)) ") . ((f | (right_open_halfline p)) . ((a . n) + (b . n)))) - (a . n) by A24, A26, FUNCT_1:47
.= ((a . n) + (b . n)) - (a . n) by A27, FUNCT_1:34
.= b . n ; ::_thesis: verum
end;
A28: h + c is convergent by A15, Th4;
then A29: ((f | (right_open_halfline p)) ") /* (h + c) is convergent by A16, A17, A14, FCONT_1:def_1;
then (((f | (right_open_halfline p)) ") /* (h + c)) - a is convergent by SEQ_2:11;
then A30: b is convergent by A25, FUNCT_2:63;
A31: lim a = a . 0 by SEQ_4:26
.= ((f | (right_open_halfline p)) ") . y0 by FUNCOP_1:7 ;
((f | (right_open_halfline p)) ") . y0 = lim (((f | (right_open_halfline p)) ") /* (h + c)) by A16, A28, A17, A14, FCONT_1:def_1;
then lim ((((f | (right_open_halfline p)) ") /* (h + c)) - a) = (((f | (right_open_halfline p)) ") . y0) - (((f | (right_open_halfline p)) ") . y0) by A29, A31, SEQ_2:12
.= 0 ;
then A32: lim b = 0 by A25, FUNCT_2:63;
A33: rng (b + a) c= dom f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (b + a) or x in dom f )
assume x in rng (b + a) ; ::_thesis: x in dom f
then consider n being Element of NAT such that
A34: x = (b + a) . n by FUNCT_2:113;
A35: (h + c) . n = (h + c) . n ;
x = (a . n) + (b . n) by A34, SEQ_1:7;
hence x in dom f by A24, A35; ::_thesis: verum
end;
((f | (right_open_halfline p)) ") . y0 in dom (f | (right_open_halfline p)) by A6, A8, FUNCT_1:def_3;
then ((f | (right_open_halfline p)) ") . y0 in (dom f) /\ (right_open_halfline p) by RELAT_1:61;
then A36: ((f | (right_open_halfline p)) ") . y0 in right_open_halfline p by XBOOLE_0:def_4;
then A37: f is_differentiable_in ((f | (right_open_halfline p)) ") . y0 by A2, FDIFF_1:9;
A38: now__::_thesis:_for_n_being_Element_of_NAT_holds_c_._n_=_(f_|_(right_open_halfline_p))_._x0
let n be Element of NAT ; ::_thesis: c . n = (f | (right_open_halfline p)) . x0
c . n in rng c by VALUED_0:28;
hence c . n = (f | (right_open_halfline p)) . x0 by A10, A15, TARSKI:def_1; ::_thesis: verum
end;
A39: 0 <> diff (f,(((f | (right_open_halfline p)) ") . y0)) by A3, A36;
now__::_thesis:_for_n_being_Element_of_NAT_holds_not_b_._n_=_0
given n being Element of NAT such that A41: b . n = 0 ; ::_thesis: contradiction
a . n = ((f | (right_open_halfline p)) ") . ((f | (right_open_halfline p)) . x0) by A10, FUNCOP_1:7
.= x0 by A9, FUNCT_1:34 ;
then A42: f . ((a . n) + (b . n)) = (f | (right_open_halfline p)) . x0 by A9, A41, FUNCT_1:47;
(h + c) . n = (h . n) + (c . n) by SEQ_1:7
.= (h . n) + ((f | (right_open_halfline p)) . x0) by A38 ;
then (h . n) + ((f | (right_open_halfline p)) . x0) = (f | (right_open_halfline p)) . x0 by A24, A42;
hence contradiction by SEQ_1:5; ::_thesis: verum
end;
then b is non-zero by SEQ_1:5;
then reconsider b = b as non-zero 0 -convergent Real_Sequence by A30, A32, FDIFF_1:def_1;
A44: b " is non-zero by SEQ_1:33;
A45: rng a = {(((f | (right_open_halfline p)) ") . y0)}
proof
thus rng a c= {(((f | (right_open_halfline p)) ") . y0)} :: according to XBOOLE_0:def_10 ::_thesis: {(((f | (right_open_halfline p)) ") . y0)} c= rng a
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng a or x in {(((f | (right_open_halfline p)) ") . y0)} )
assume x in rng a ; ::_thesis: x in {(((f | (right_open_halfline p)) ") . y0)}
then ex n being Element of NAT st x = a . n by FUNCT_2:113;
then x = ((f | (right_open_halfline p)) ") . y0 by FUNCOP_1:7;
hence x in {(((f | (right_open_halfline p)) ") . y0)} by TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(((f | (right_open_halfline p)) ") . y0)} or x in rng a )
assume x in {(((f | (right_open_halfline p)) ") . y0)} ; ::_thesis: x in rng a
then x = ((f | (right_open_halfline p)) ") . y0 by TARSKI:def_1;
then a . 0 = x by FUNCOP_1:7;
hence x in rng a by VALUED_0:28; ::_thesis: verum
end;
A46: rng a c= dom f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng a or x in dom f )
assume x in rng a ; ::_thesis: x in dom f
then x = ((f | (right_open_halfline p)) ") . y0 by A45, TARSKI:def_1;
then x = x0 by A9, A10, FUNCT_1:34;
then x in (dom f) /\ (right_open_halfline p) by A9, RELAT_1:61;
hence x in dom f by XBOOLE_0:def_4; ::_thesis: verum
end;
now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_a)_._n_=_c_._n
let n be Element of NAT ; ::_thesis: (f /* a) . n = c . n
A47: ((f | (right_open_halfline p)) ") . y0 in rng ((f | (right_open_halfline p)) ") by A8, FUNCT_1:def_3;
c . n in rng c by VALUED_0:28;
then A48: c . n = y0 by A15, TARSKI:def_1;
thus (f /* a) . n = f . (a . n) by A46, FUNCT_2:108
.= f . (((f | (right_open_halfline p)) ") . y0) by FUNCOP_1:7
.= (f | (right_open_halfline p)) . (((f | (right_open_halfline p)) ") . y0) by A6, A47, FUNCT_1:47
.= c . n by A5, A8, A48, FUNCT_1:35 ; ::_thesis: verum
end;
then A49: f /* a = c by FUNCT_2:63;
now__::_thesis:_for_n_being_Element_of_NAT_holds_h_._n_=_((f_/*_(b_+_a))_-_(f_/*_a))_._n
let n be Element of NAT ; ::_thesis: h . n = ((f /* (b + a)) - (f /* a)) . n
(h + c) . n = f . ((a . n) + (b . n)) by A24;
then (h . n) + (c . n) = f . ((a . n) + (b . n)) by SEQ_1:7;
hence h . n = (f . ((b . n) + (a . n))) - ((f /* a) . n) by A49
.= (f . ((b + a) . n)) - ((f /* a) . n) by SEQ_1:7
.= ((f /* (b + a)) . n) - ((f /* a) . n) by A33, FUNCT_2:108
.= ((f /* (b + a)) - (f /* a)) . n by RFUNCT_2:1 ;
::_thesis: verum
end;
then A50: h = (f /* (b + a)) - (f /* a) by FUNCT_2:63;
then (f /* (b + a)) - (f /* a) is non-zero ;
then A51: (b ") (#) ((f /* (b + a)) - (f /* a)) is non-zero by A44, SEQ_1:35;
A52: rng c c= dom ((f | (right_open_halfline p)) ")
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng c or x in dom ((f | (right_open_halfline p)) ") )
assume x in rng c ; ::_thesis: x in dom ((f | (right_open_halfline p)) ")
hence x in dom ((f | (right_open_halfline p)) ") by A8, A15, TARSKI:def_1; ::_thesis: verum
end;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((h_")_(#)_((((f_|_(right_open_halfline_p))_")_/*_(h_+_c))_-_(((f_|_(right_open_halfline_p))_")_/*_c)))_._n_=_(((b_")_(#)_((f_/*_(b_+_a))_-_(f_/*_a)))_")_._n
let n be Element of NAT ; ::_thesis: ((h ") (#) ((((f | (right_open_halfline p)) ") /* (h + c)) - (((f | (right_open_halfline p)) ") /* c))) . n = (((b ") (#) ((f /* (b + a)) - (f /* a))) ") . n
A53: (h + c) . n = (h + c) . n ;
then A54: (a . n) + (b . n) in dom (f | (right_open_halfline p)) by A24;
c . n in rng c by VALUED_0:28;
then A55: c . n = y0 by A15, TARSKI:def_1;
thus ((h ") (#) ((((f | (right_open_halfline p)) ") /* (h + c)) - (((f | (right_open_halfline p)) ") /* c))) . n = ((h ") . n) * (((((f | (right_open_halfline p)) ") /* (h + c)) - (((f | (right_open_halfline p)) ") /* c)) . n) by SEQ_1:8
.= ((h ") . n) * (((((f | (right_open_halfline p)) ") /* (h + c)) . n) - ((((f | (right_open_halfline p)) ") /* c) . n)) by RFUNCT_2:1
.= ((h ") . n) * ((((f | (right_open_halfline p)) ") . ((h + c) . n)) - ((((f | (right_open_halfline p)) ") /* c) . n)) by A16, FUNCT_2:108
.= ((h ") . n) * ((((f | (right_open_halfline p)) ") . (f . ((a . n) + (b . n)))) - ((((f | (right_open_halfline p)) ") /* c) . n)) by A24
.= ((h ") . n) * ((((f | (right_open_halfline p)) ") . ((f | (right_open_halfline p)) . ((a . n) + (b . n)))) - ((((f | (right_open_halfline p)) ") /* c) . n)) by A24, A53, FUNCT_1:47
.= ((h ") . n) * (((a . n) + (b . n)) - ((((f | (right_open_halfline p)) ") /* c) . n)) by A54, FUNCT_1:34
.= ((h ") . n) * (((a . n) + (b . n)) - (((f | (right_open_halfline p)) ") . (c . n))) by A52, FUNCT_2:108
.= ((h ") . n) * (((a . n) + (b . n)) - (a . n)) by A55, FUNCOP_1:7
.= ((h ") (#) ((b ") ")) . n by SEQ_1:8
.= (((b ") (#) ((f /* (b + a)) - (f /* a))) ") . n by A50, SEQ_1:36 ; ::_thesis: verum
end;
then A56: (h ") (#) ((((f | (right_open_halfline p)) ") /* (h + c)) - (((f | (right_open_halfline p)) ") /* c)) = ((b ") (#) ((f /* (b + a)) - (f /* a))) " by FUNCT_2:63;
diff (f,(((f | (right_open_halfline p)) ") . y0)) = diff (f,(((f | (right_open_halfline p)) ") . y0)) ;
then A57: (b ") (#) ((f /* (b + a)) - (f /* a)) is convergent by A45, A33, A37, Th12;
A58: lim ((b ") (#) ((f /* (b + a)) - (f /* a))) = diff (f,(((f | (right_open_halfline p)) ") . y0)) by A45, A33, A37, Th12;
hence (h ") (#) ((((f | (right_open_halfline p)) ") /* (h + c)) - (((f | (right_open_halfline p)) ") /* c)) is convergent by A56, A51, A57, A39, SEQ_2:21; ::_thesis: lim ((h ") (#) ((((f | (right_open_halfline p)) ") /* (h + c)) - (((f | (right_open_halfline p)) ") /* c))) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . y0)))
thus lim ((h ") (#) ((((f | (right_open_halfline p)) ") /* (h + c)) - (((f | (right_open_halfline p)) ") /* c))) = (diff (f,(((f | (right_open_halfline p)) ") . y0))) " by A56, A51, A57, A58, A39, SEQ_2:22
.= 1 / (diff (f,(((f | (right_open_halfline p)) ") . y0))) by XCMPLX_1:215 ; ::_thesis: verum
end;
ex N being Neighbourhood of y0 st N c= dom ((f | (right_open_halfline p)) ") by A4, A5, A8, RCOMP_1:18;
hence ( (f | (right_open_halfline p)) " is_differentiable_in y0 & diff (((f | (right_open_halfline p)) "),y0) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . y0))) ) by A11, Th12; ::_thesis: verum
end;
then for y0 being Real st y0 in dom ((f | (right_open_halfline p)) ") holds
(f | (right_open_halfline p)) " is_differentiable_in y0 ;
hence (f | (right_open_halfline p)) " is_differentiable_on dom ((f | (right_open_halfline p)) ") by A4, A5, FDIFF_1:9; ::_thesis: for x0 being Real st x0 in dom ((f | (right_open_halfline p)) ") holds
diff (((f | (right_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . x0)))
let x0 be Real; ::_thesis: ( x0 in dom ((f | (right_open_halfline p)) ") implies diff (((f | (right_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . x0))) )
assume x0 in dom ((f | (right_open_halfline p)) ") ; ::_thesis: diff (((f | (right_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . x0)))
hence diff (((f | (right_open_halfline p)) "),x0) = 1 / (diff (f,(((f | (right_open_halfline p)) ") . x0))) by A7; ::_thesis: verum
end;
theorem :: FDIFF_2:48
for p, g being Real
for f being one-to-one PartFunc of REAL,REAL st ].p,g.[ c= dom f & f is_differentiable_on ].p,g.[ & ( for x0 being Real st x0 in ].p,g.[ holds
0 < diff (f,x0) or for x0 being Real st x0 in ].p,g.[ holds
diff (f,x0) < 0 ) holds
( f | ].p,g.[ is one-to-one & (f | ].p,g.[) " is_differentiable_on dom ((f | ].p,g.[) ") & ( for x0 being Real st x0 in dom ((f | ].p,g.[) ") holds
diff (((f | ].p,g.[) "),x0) = 1 / (diff (f,(((f | ].p,g.[) ") . x0))) ) )
proof
let p, g be Real; ::_thesis: for f being one-to-one PartFunc of REAL,REAL st ].p,g.[ c= dom f & f is_differentiable_on ].p,g.[ & ( for x0 being Real st x0 in ].p,g.[ holds
0 < diff (f,x0) or for x0 being Real st x0 in ].p,g.[ holds
diff (f,x0) < 0 ) holds
( f | ].p,g.[ is one-to-one & (f | ].p,g.[) " is_differentiable_on dom ((f | ].p,g.[) ") & ( for x0 being Real st x0 in dom ((f | ].p,g.[) ") holds
diff (((f | ].p,g.[) "),x0) = 1 / (diff (f,(((f | ].p,g.[) ") . x0))) ) )
let f be one-to-one PartFunc of REAL,REAL; ::_thesis: ( ].p,g.[ c= dom f & f is_differentiable_on ].p,g.[ & ( for x0 being Real st x0 in ].p,g.[ holds
0 < diff (f,x0) or for x0 being Real st x0 in ].p,g.[ holds
diff (f,x0) < 0 ) implies ( f | ].p,g.[ is one-to-one & (f | ].p,g.[) " is_differentiable_on dom ((f | ].p,g.[) ") & ( for x0 being Real st x0 in dom ((f | ].p,g.[) ") holds
diff (((f | ].p,g.[) "),x0) = 1 / (diff (f,(((f | ].p,g.[) ") . x0))) ) ) )
set l = ].p,g.[;
assume that
A1: ].p,g.[ c= dom f and
A2: f is_differentiable_on ].p,g.[ and
A3: ( for x0 being Real st x0 in ].p,g.[ holds
0 < diff (f,x0) or for x0 being Real st x0 in ].p,g.[ holds
diff (f,x0) < 0 ) ; ::_thesis: ( f | ].p,g.[ is one-to-one & (f | ].p,g.[) " is_differentiable_on dom ((f | ].p,g.[) ") & ( for x0 being Real st x0 in dom ((f | ].p,g.[) ") holds
diff (((f | ].p,g.[) "),x0) = 1 / (diff (f,(((f | ].p,g.[) ") . x0))) ) )
A4: rng (f | ].p,g.[) is open by A1, A2, A3, Th41;
set f1 = f | ].p,g.[;
thus f | ].p,g.[ is one-to-one ; ::_thesis: ( (f | ].p,g.[) " is_differentiable_on dom ((f | ].p,g.[) ") & ( for x0 being Real st x0 in dom ((f | ].p,g.[) ") holds
diff (((f | ].p,g.[) "),x0) = 1 / (diff (f,(((f | ].p,g.[) ") . x0))) ) )
A5: dom ((f | ].p,g.[) ") = rng (f | ].p,g.[) by FUNCT_1:33;
A6: rng ((f | ].p,g.[) ") = dom (f | ].p,g.[) by FUNCT_1:33;
A7: for y0 being Real st y0 in dom ((f | ].p,g.[) ") holds
( (f | ].p,g.[) " is_differentiable_in y0 & diff (((f | ].p,g.[) "),y0) = 1 / (diff (f,(((f | ].p,g.[) ") . y0))) )
proof
let y0 be Real; ::_thesis: ( y0 in dom ((f | ].p,g.[) ") implies ( (f | ].p,g.[) " is_differentiable_in y0 & diff (((f | ].p,g.[) "),y0) = 1 / (diff (f,(((f | ].p,g.[) ") . y0))) ) )
assume A8: y0 in dom ((f | ].p,g.[) ") ; ::_thesis: ( (f | ].p,g.[) " is_differentiable_in y0 & diff (((f | ].p,g.[) "),y0) = 1 / (diff (f,(((f | ].p,g.[) ") . y0))) )
then consider x0 being Real such that
A9: x0 in dom (f | ].p,g.[) and
A10: y0 = (f | ].p,g.[) . x0 by A5, PARTFUN1:3;
A11: for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {y0} & rng (h + c) c= dom ((f | ].p,g.[) ") holds
( (h ") (#) ((((f | ].p,g.[) ") /* (h + c)) - (((f | ].p,g.[) ") /* c)) is convergent & lim ((h ") (#) ((((f | ].p,g.[) ") /* (h + c)) - (((f | ].p,g.[) ") /* c))) = 1 / (diff (f,(((f | ].p,g.[) ") . y0))) )
proof
A12: ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing )
proof
percases ( for x0 being Real st x0 in ].p,g.[ holds
0 < diff (f,x0) or for x0 being Real st x0 in ].p,g.[ holds
diff (f,x0) < 0 ) by A3;
suppose for x0 being Real st x0 in ].p,g.[ holds
0 < diff (f,x0) ; ::_thesis: ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing )
hence ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing ) by A1, A2, ROLLE:9; ::_thesis: verum
end;
suppose for x0 being Real st x0 in ].p,g.[ holds
diff (f,x0) < 0 ; ::_thesis: ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing )
hence ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing ) by A1, A2, ROLLE:10; ::_thesis: verum
end;
end;
end;
].p,g.[ c= dom f by A2, FDIFF_1:def_6;
then ((f | ].p,g.[) ") | (f .: ].p,g.[) is continuous by A12, FCONT_3:17;
then A13: ((f | ].p,g.[) ") | (rng (f | ].p,g.[)) is continuous by RELAT_1:115;
y0 in dom (((f | ].p,g.[) ") | (rng (f | ].p,g.[))) by A5, A8, RELAT_1:69;
then ((f | ].p,g.[) ") | (rng (f | ].p,g.[)) is_continuous_in y0 by A13, FCONT_1:def_2;
then A14: (f | ].p,g.[) " is_continuous_in y0 by A5, RELAT_1:68;
reconsider a = NAT --> (((f | ].p,g.[) ") . y0) as Real_Sequence by FUNCOP_1:45;
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: for c being V8() Real_Sequence st rng c = {y0} & rng (h + c) c= dom ((f | ].p,g.[) ") holds
( (h ") (#) ((((f | ].p,g.[) ") /* (h + c)) - (((f | ].p,g.[) ") /* c)) is convergent & lim ((h ") (#) ((((f | ].p,g.[) ") /* (h + c)) - (((f | ].p,g.[) ") /* c))) = 1 / (diff (f,(((f | ].p,g.[) ") . y0))) )
let c be V8() Real_Sequence; ::_thesis: ( rng c = {y0} & rng (h + c) c= dom ((f | ].p,g.[) ") implies ( (h ") (#) ((((f | ].p,g.[) ") /* (h + c)) - (((f | ].p,g.[) ") /* c)) is convergent & lim ((h ") (#) ((((f | ].p,g.[) ") /* (h + c)) - (((f | ].p,g.[) ") /* c))) = 1 / (diff (f,(((f | ].p,g.[) ") . y0))) ) )
assume that
A15: rng c = {y0} and
A16: rng (h + c) c= dom ((f | ].p,g.[) ") ; ::_thesis: ( (h ") (#) ((((f | ].p,g.[) ") /* (h + c)) - (((f | ].p,g.[) ") /* c)) is convergent & lim ((h ") (#) ((((f | ].p,g.[) ") /* (h + c)) - (((f | ].p,g.[) ") /* c))) = 1 / (diff (f,(((f | ].p,g.[) ") . y0))) )
A17: lim (h + c) = y0 by A15, Th4;
reconsider a = a as V8() Real_Sequence ;
defpred S1[ Element of NAT , real number ] means for r1, r2 being real number st r1 = (h + c) . $1 & r2 = a . $1 holds
( r1 = f . (r2 + $2) & r2 + $2 in dom f & r2 + $2 in dom (f | ].p,g.[) );
A18: for n being Element of NAT ex r being Real st S1[n,r]
proof
let n be Element of NAT ; ::_thesis: ex r being Real st S1[n,r]
(h + c) . n in rng (h + c) by VALUED_0:28;
then consider g being Real such that
A19: g in dom (f | ].p,g.[) and
A20: (h + c) . n = (f | ].p,g.[) . g by A5, A16, PARTFUN1:3;
take r = g - x0; ::_thesis: S1[n,r]
let r1, r2 be real number ; ::_thesis: ( r1 = (h + c) . n & r2 = a . n implies ( r1 = f . (r2 + r) & r2 + r in dom f & r2 + r in dom (f | ].p,g.[) ) )
assume that
A21: r1 = (h + c) . n and
A22: r2 = a . n ; ::_thesis: ( r1 = f . (r2 + r) & r2 + r in dom f & r2 + r in dom (f | ].p,g.[) )
A23: a . n = ((f | ].p,g.[) ") . ((f | ].p,g.[) . x0) by A10, FUNCOP_1:7
.= x0 by A9, FUNCT_1:34 ;
hence r1 = f . (r2 + r) by A19, A20, A21, A22, FUNCT_1:47; ::_thesis: ( r2 + r in dom f & r2 + r in dom (f | ].p,g.[) )
g in (dom f) /\ ].p,g.[ by A19, RELAT_1:61;
hence ( r2 + r in dom f & r2 + r in dom (f | ].p,g.[) ) by A19, A23, A22, XBOOLE_0:def_4; ::_thesis: verum
end;
consider b being Real_Sequence such that
A24: for n being Element of NAT holds S1[n,b . n] from FUNCT_2:sch_3(A18);
A25: now__::_thesis:_for_n_being_Element_of_NAT_holds_((((f_|_].p,g.[)_")_/*_(h_+_c))_-_a)_._n_=_b_._n
let n be Element of NAT ; ::_thesis: ((((f | ].p,g.[) ") /* (h + c)) - a) . n = b . n
A26: (h + c) . n = (h + c) . n ;
then A27: (a . n) + (b . n) in dom (f | ].p,g.[) by A24;
thus ((((f | ].p,g.[) ") /* (h + c)) - a) . n = ((((f | ].p,g.[) ") /* (h + c)) . n) - (a . n) by RFUNCT_2:1
.= (((f | ].p,g.[) ") . ((h + c) . n)) - (a . n) by A16, FUNCT_2:108
.= (((f | ].p,g.[) ") . (f . ((a . n) + (b . n)))) - (a . n) by A24
.= (((f | ].p,g.[) ") . ((f | ].p,g.[) . ((a . n) + (b . n)))) - (a . n) by A24, A26, FUNCT_1:47
.= ((a . n) + (b . n)) - (a . n) by A27, FUNCT_1:34
.= b . n ; ::_thesis: verum
end;
A28: h + c is convergent by A15, Th4;
then A29: ((f | ].p,g.[) ") /* (h + c) is convergent by A16, A17, A14, FCONT_1:def_1;
then (((f | ].p,g.[) ") /* (h + c)) - a is convergent by SEQ_2:11;
then A30: b is convergent by A25, FUNCT_2:63;
A31: lim a = a . 0 by SEQ_4:26
.= ((f | ].p,g.[) ") . y0 by FUNCOP_1:7 ;
((f | ].p,g.[) ") . y0 = lim (((f | ].p,g.[) ") /* (h + c)) by A16, A28, A17, A14, FCONT_1:def_1;
then lim ((((f | ].p,g.[) ") /* (h + c)) - a) = (((f | ].p,g.[) ") . y0) - (((f | ].p,g.[) ") . y0) by A29, A31, SEQ_2:12
.= 0 ;
then A32: lim b = 0 by A25, FUNCT_2:63;
A33: rng (b + a) c= dom f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (b + a) or x in dom f )
assume x in rng (b + a) ; ::_thesis: x in dom f
then consider n being Element of NAT such that
A34: x = (b + a) . n by FUNCT_2:113;
A35: (h + c) . n = (h + c) . n ;
x = (a . n) + (b . n) by A34, SEQ_1:7;
hence x in dom f by A24, A35; ::_thesis: verum
end;
((f | ].p,g.[) ") . y0 in dom (f | ].p,g.[) by A6, A8, FUNCT_1:def_3;
then ((f | ].p,g.[) ") . y0 in (dom f) /\ ].p,g.[ by RELAT_1:61;
then A36: ((f | ].p,g.[) ") . y0 in ].p,g.[ by XBOOLE_0:def_4;
then A37: f is_differentiable_in ((f | ].p,g.[) ") . y0 by A2, FDIFF_1:9;
A38: now__::_thesis:_for_n_being_Element_of_NAT_holds_c_._n_=_(f_|_].p,g.[)_._x0
let n be Element of NAT ; ::_thesis: c . n = (f | ].p,g.[) . x0
c . n in rng c by VALUED_0:28;
hence c . n = (f | ].p,g.[) . x0 by A10, A15, TARSKI:def_1; ::_thesis: verum
end;
A39: 0 <> diff (f,(((f | ].p,g.[) ") . y0)) by A3, A36;
now__::_thesis:_for_n_being_Element_of_NAT_holds_not_b_._n_=_0
given n being Element of NAT such that A41: b . n = 0 ; ::_thesis: contradiction
a . n = ((f | ].p,g.[) ") . ((f | ].p,g.[) . x0) by A10, FUNCOP_1:7
.= x0 by A9, FUNCT_1:34 ;
then A42: f . ((a . n) + (b . n)) = (f | ].p,g.[) . x0 by A9, A41, FUNCT_1:47;
(h + c) . n = (h . n) + (c . n) by SEQ_1:7
.= (h . n) + ((f | ].p,g.[) . x0) by A38 ;
then (h . n) + ((f | ].p,g.[) . x0) = (f | ].p,g.[) . x0 by A24, A42;
hence contradiction by SEQ_1:5; ::_thesis: verum
end;
then b is non-zero by SEQ_1:5;
then reconsider b = b as non-zero 0 -convergent Real_Sequence by A30, A32, FDIFF_1:def_1;
A44: b " is non-zero by SEQ_1:33;
A45: rng a = {(((f | ].p,g.[) ") . y0)}
proof
thus rng a c= {(((f | ].p,g.[) ") . y0)} :: according to XBOOLE_0:def_10 ::_thesis: {(((f | ].p,g.[) ") . y0)} c= rng a
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng a or x in {(((f | ].p,g.[) ") . y0)} )
assume x in rng a ; ::_thesis: x in {(((f | ].p,g.[) ") . y0)}
then ex n being Element of NAT st x = a . n by FUNCT_2:113;
then x = ((f | ].p,g.[) ") . y0 by FUNCOP_1:7;
hence x in {(((f | ].p,g.[) ") . y0)} by TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(((f | ].p,g.[) ") . y0)} or x in rng a )
assume x in {(((f | ].p,g.[) ") . y0)} ; ::_thesis: x in rng a
then x = ((f | ].p,g.[) ") . y0 by TARSKI:def_1;
then a . 0 = x by FUNCOP_1:7;
hence x in rng a by VALUED_0:28; ::_thesis: verum
end;
A46: rng a c= dom f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng a or x in dom f )
assume x in rng a ; ::_thesis: x in dom f
then x = ((f | ].p,g.[) ") . y0 by A45, TARSKI:def_1;
then x = x0 by A9, A10, FUNCT_1:34;
then x in (dom f) /\ ].p,g.[ by A9, RELAT_1:61;
hence x in dom f by XBOOLE_0:def_4; ::_thesis: verum
end;
now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_a)_._n_=_c_._n
let n be Element of NAT ; ::_thesis: (f /* a) . n = c . n
A47: ((f | ].p,g.[) ") . y0 in rng ((f | ].p,g.[) ") by A8, FUNCT_1:def_3;
c . n in rng c by VALUED_0:28;
then A48: c . n = y0 by A15, TARSKI:def_1;
thus (f /* a) . n = f . (a . n) by A46, FUNCT_2:108
.= f . (((f | ].p,g.[) ") . y0) by FUNCOP_1:7
.= (f | ].p,g.[) . (((f | ].p,g.[) ") . y0) by A6, A47, FUNCT_1:47
.= c . n by A5, A8, A48, FUNCT_1:35 ; ::_thesis: verum
end;
then A49: f /* a = c by FUNCT_2:63;
now__::_thesis:_for_n_being_Element_of_NAT_holds_h_._n_=_((f_/*_(b_+_a))_-_(f_/*_a))_._n
let n be Element of NAT ; ::_thesis: h . n = ((f /* (b + a)) - (f /* a)) . n
(h + c) . n = f . ((a . n) + (b . n)) by A24;
then (h . n) + (c . n) = f . ((a . n) + (b . n)) by SEQ_1:7;
hence h . n = (f . ((b . n) + (a . n))) - ((f /* a) . n) by A49
.= (f . ((b + a) . n)) - ((f /* a) . n) by SEQ_1:7
.= ((f /* (b + a)) . n) - ((f /* a) . n) by A33, FUNCT_2:108
.= ((f /* (b + a)) - (f /* a)) . n by RFUNCT_2:1 ;
::_thesis: verum
end;
then A50: h = (f /* (b + a)) - (f /* a) by FUNCT_2:63;
then (f /* (b + a)) - (f /* a) is non-zero ;
then A51: (b ") (#) ((f /* (b + a)) - (f /* a)) is non-zero by A44, SEQ_1:35;
A52: rng c c= dom ((f | ].p,g.[) ")
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng c or x in dom ((f | ].p,g.[) ") )
assume x in rng c ; ::_thesis: x in dom ((f | ].p,g.[) ")
hence x in dom ((f | ].p,g.[) ") by A8, A15, TARSKI:def_1; ::_thesis: verum
end;
now__::_thesis:_for_n_being_Element_of_NAT_holds_((h_")_(#)_((((f_|_].p,g.[)_")_/*_(h_+_c))_-_(((f_|_].p,g.[)_")_/*_c)))_._n_=_(((b_")_(#)_((f_/*_(b_+_a))_-_(f_/*_a)))_")_._n
let n be Element of NAT ; ::_thesis: ((h ") (#) ((((f | ].p,g.[) ") /* (h + c)) - (((f | ].p,g.[) ") /* c))) . n = (((b ") (#) ((f /* (b + a)) - (f /* a))) ") . n
A53: (h + c) . n = (h + c) . n ;
then A54: (a . n) + (b . n) in dom (f | ].p,g.[) by A24;
c . n in rng c by VALUED_0:28;
then A55: c . n = y0 by A15, TARSKI:def_1;
thus ((h ") (#) ((((f | ].p,g.[) ") /* (h + c)) - (((f | ].p,g.[) ") /* c))) . n = ((h ") . n) * (((((f | ].p,g.[) ") /* (h + c)) - (((f | ].p,g.[) ") /* c)) . n) by SEQ_1:8
.= ((h ") . n) * (((((f | ].p,g.[) ") /* (h + c)) . n) - ((((f | ].p,g.[) ") /* c) . n)) by RFUNCT_2:1
.= ((h ") . n) * ((((f | ].p,g.[) ") . ((h + c) . n)) - ((((f | ].p,g.[) ") /* c) . n)) by A16, FUNCT_2:108
.= ((h ") . n) * ((((f | ].p,g.[) ") . (f . ((a . n) + (b . n)))) - ((((f | ].p,g.[) ") /* c) . n)) by A24
.= ((h ") . n) * ((((f | ].p,g.[) ") . ((f | ].p,g.[) . ((a . n) + (b . n)))) - ((((f | ].p,g.[) ") /* c) . n)) by A24, A53, FUNCT_1:47
.= ((h ") . n) * (((a . n) + (b . n)) - ((((f | ].p,g.[) ") /* c) . n)) by A54, FUNCT_1:34
.= ((h ") . n) * (((a . n) + (b . n)) - (((f | ].p,g.[) ") . (c . n))) by A52, FUNCT_2:108
.= ((h ") . n) * (((a . n) + (b . n)) - (a . n)) by A55, FUNCOP_1:7
.= ((h ") (#) ((b ") ")) . n by SEQ_1:8
.= (((b ") (#) ((f /* (b + a)) - (f /* a))) ") . n by A50, SEQ_1:36 ; ::_thesis: verum
end;
then A56: (h ") (#) ((((f | ].p,g.[) ") /* (h + c)) - (((f | ].p,g.[) ") /* c)) = ((b ") (#) ((f /* (b + a)) - (f /* a))) " by FUNCT_2:63;
diff (f,(((f | ].p,g.[) ") . y0)) = diff (f,(((f | ].p,g.[) ") . y0)) ;
then A57: (b ") (#) ((f /* (b + a)) - (f /* a)) is convergent by A45, A33, A37, Th12;
A58: lim ((b ") (#) ((f /* (b + a)) - (f /* a))) = diff (f,(((f | ].p,g.[) ") . y0)) by A45, A33, A37, Th12;
hence (h ") (#) ((((f | ].p,g.[) ") /* (h + c)) - (((f | ].p,g.[) ") /* c)) is convergent by A56, A51, A57, A39, SEQ_2:21; ::_thesis: lim ((h ") (#) ((((f | ].p,g.[) ") /* (h + c)) - (((f | ].p,g.[) ") /* c))) = 1 / (diff (f,(((f | ].p,g.[) ") . y0)))
thus lim ((h ") (#) ((((f | ].p,g.[) ") /* (h + c)) - (((f | ].p,g.[) ") /* c))) = (diff (f,(((f | ].p,g.[) ") . y0))) " by A56, A51, A57, A58, A39, SEQ_2:22
.= 1 / (diff (f,(((f | ].p,g.[) ") . y0))) by XCMPLX_1:215 ; ::_thesis: verum
end;
ex N being Neighbourhood of y0 st N c= dom ((f | ].p,g.[) ") by A4, A5, A8, RCOMP_1:18;
hence ( (f | ].p,g.[) " is_differentiable_in y0 & diff (((f | ].p,g.[) "),y0) = 1 / (diff (f,(((f | ].p,g.[) ") . y0))) ) by A11, Th12; ::_thesis: verum
end;
then for y0 being Real st y0 in dom ((f | ].p,g.[) ") holds
(f | ].p,g.[) " is_differentiable_in y0 ;
hence (f | ].p,g.[) " is_differentiable_on dom ((f | ].p,g.[) ") by A4, A5, FDIFF_1:9; ::_thesis: for x0 being Real st x0 in dom ((f | ].p,g.[) ") holds
diff (((f | ].p,g.[) "),x0) = 1 / (diff (f,(((f | ].p,g.[) ") . x0)))
let x0 be Real; ::_thesis: ( x0 in dom ((f | ].p,g.[) ") implies diff (((f | ].p,g.[) "),x0) = 1 / (diff (f,(((f | ].p,g.[) ") . x0))) )
assume x0 in dom ((f | ].p,g.[) ") ; ::_thesis: diff (((f | ].p,g.[) "),x0) = 1 / (diff (f,(((f | ].p,g.[) ") . x0)))
hence diff (((f | ].p,g.[) "),x0) = 1 / (diff (f,(((f | ].p,g.[) ") . x0))) by A7; ::_thesis: verum
end;
theorem :: FDIFF_2:49
for x0 being Real
for f being PartFunc of REAL,REAL st f is_differentiable_in x0 holds
for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f & rng ((- h) + c) c= dom f holds
( ((2 (#) h) ") (#) ((f /* (c + h)) - (f /* (c - h))) is convergent & lim (((2 (#) h) ") (#) ((f /* (c + h)) - (f /* (c - h)))) = diff (f,x0) )
proof
let x0 be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_differentiable_in x0 holds
for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f & rng ((- h) + c) c= dom f holds
( ((2 (#) h) ") (#) ((f /* (c + h)) - (f /* (c - h))) is convergent & lim (((2 (#) h) ") (#) ((f /* (c + h)) - (f /* (c - h)))) = diff (f,x0) )
let f be PartFunc of REAL,REAL; ::_thesis: ( f is_differentiable_in x0 implies for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f & rng ((- h) + c) c= dom f holds
( ((2 (#) h) ") (#) ((f /* (c + h)) - (f /* (c - h))) is convergent & lim (((2 (#) h) ") (#) ((f /* (c + h)) - (f /* (c - h)))) = diff (f,x0) ) )
assume A1: f is_differentiable_in x0 ; ::_thesis: for h being non-zero 0 -convergent Real_Sequence
for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f & rng ((- h) + c) c= dom f holds
( ((2 (#) h) ") (#) ((f /* (c + h)) - (f /* (c - h))) is convergent & lim (((2 (#) h) ") (#) ((f /* (c + h)) - (f /* (c - h)))) = diff (f,x0) )
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: for c being V8() Real_Sequence st rng c = {x0} & rng (h + c) c= dom f & rng ((- h) + c) c= dom f holds
( ((2 (#) h) ") (#) ((f /* (c + h)) - (f /* (c - h))) is convergent & lim (((2 (#) h) ") (#) ((f /* (c + h)) - (f /* (c - h)))) = diff (f,x0) )
let c be V8() Real_Sequence; ::_thesis: ( rng c = {x0} & rng (h + c) c= dom f & rng ((- h) + c) c= dom f implies ( ((2 (#) h) ") (#) ((f /* (c + h)) - (f /* (c - h))) is convergent & lim (((2 (#) h) ") (#) ((f /* (c + h)) - (f /* (c - h)))) = diff (f,x0) ) )
assume that
A2: rng c = {x0} and
A3: rng (h + c) c= dom f and
A4: rng ((- h) + c) c= dom f ; ::_thesis: ( ((2 (#) h) ") (#) ((f /* (c + h)) - (f /* (c - h))) is convergent & lim (((2 (#) h) ") (#) ((f /* (c + h)) - (f /* (c - h)))) = diff (f,x0) )
set fm = ((- h) ") (#) ((f /* ((- h) + c)) - (f /* c));
lim (- h) = - (lim h) by SEQ_2:10;
then A: - h is 0 -convergent by FDIFF_1:def_1;
A5: lim (((- h) ") (#) ((f /* ((- h) + c)) - (f /* c))) = diff (f,x0) by A1, A2, A4, Th12, A;
set fp = (h ") (#) ((f /* (h + c)) - (f /* c));
A6: diff (f,x0) = diff (f,x0) ;
then A7: ((- h) ") (#) ((f /* ((- h) + c)) - (f /* c)) is convergent by A1, A2, A4, Th12, A;
A8: (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent by A1, A2, A3, A6, Th12;
then A9: ((h ") (#) ((f /* (h + c)) - (f /* c))) + (((- h) ") (#) ((f /* ((- h) + c)) - (f /* c))) is convergent by A7, SEQ_2:5;
A10: now__::_thesis:_for_n_being_Element_of_NAT_holds_(((f_/*_(c_+_h))_-_(f_/*_(c_-_h)))_+_((f_/*_c)_-_(f_/*_c)))_._n_=_((f_/*_(c_+_h))_-_(f_/*_(c_-_h)))_._n
let n be Element of NAT ; ::_thesis: (((f /* (c + h)) - (f /* (c - h))) + ((f /* c) - (f /* c))) . n = ((f /* (c + h)) - (f /* (c - h))) . n
thus (((f /* (c + h)) - (f /* (c - h))) + ((f /* c) - (f /* c))) . n = (((f /* (c + h)) - (f /* (c - h))) . n) + (((f /* c) - (f /* c)) . n) by SEQ_1:7
.= (((f /* (c + h)) - (f /* (c - h))) . n) + (((f /* c) . n) - ((f /* c) . n)) by RFUNCT_2:1
.= ((f /* (c + h)) - (f /* (c - h))) . n ; ::_thesis: verum
end;
A11: (2 ") (#) (((h ") (#) ((f /* (h + c)) - (f /* c))) + (((- h) ") (#) ((f /* ((- h) + c)) - (f /* c)))) = (2 ") (#) (((h ") (#) ((f /* (c + h)) - (f /* c))) + (((- 1) (#) (h ")) (#) ((f /* (c + (- h))) - (f /* c)))) by SEQ_1:47
.= (2 ") (#) (((h ") (#) ((f /* (c + h)) - (f /* c))) + ((- 1) (#) ((h ") (#) ((f /* (c + (- h))) - (f /* c))))) by SEQ_1:18
.= (2 ") (#) (((h ") (#) ((f /* (c + h)) - (f /* c))) + ((h ") (#) ((- 1) (#) ((f /* (c + (- h))) - (f /* c))))) by SEQ_1:19
.= (2 ") (#) ((h ") (#) (((f /* (c + h)) - (f /* c)) + ((- 1) (#) ((f /* (c + (- h))) - (f /* c))))) by SEQ_1:16
.= ((2 ") (#) (h ")) (#) (((f /* (c + h)) - (f /* c)) + ((- 1) (#) ((f /* (c + (- h))) - (f /* c)))) by SEQ_1:18
.= ((2 (#) h) ") (#) (((f /* (c + h)) - (f /* c)) + ((- 1) (#) ((f /* (c + (- h))) - (f /* c)))) by SEQ_1:46
.= ((2 (#) h) ") (#) ((f /* (c + h)) - ((f /* c) - (- ((f /* (c + (- h))) - (f /* c))))) by SEQ_1:30
.= ((2 (#) h) ") (#) ((f /* (c + h)) - ((f /* (c + (- h))) - ((f /* c) - (f /* c)))) by SEQ_1:30
.= ((2 (#) h) ") (#) (((f /* (c + h)) - (f /* (c - h))) + ((f /* c) - (f /* c))) by SEQ_1:30 ;
lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = diff (f,x0) by A1, A2, A3, Th12;
then lim (((h ") (#) ((f /* (h + c)) - (f /* c))) + (((- h) ") (#) ((f /* ((- h) + c)) - (f /* c)))) = (1 * (diff (f,x0))) + (diff (f,x0)) by A8, A7, A5, SEQ_2:6
.= 2 * (diff (f,x0)) ;
then A12: lim ((2 ") (#) (((h ") (#) ((f /* (h + c)) - (f /* c))) + (((- h) ") (#) ((f /* ((- h) + c)) - (f /* c))))) = (2 ") * (2 * (diff (f,x0))) by A9, SEQ_2:8
.= diff (f,x0) ;
(2 ") (#) (((h ") (#) ((f /* (h + c)) - (f /* c))) + (((- h) ") (#) ((f /* ((- h) + c)) - (f /* c)))) is convergent by A9, SEQ_2:7;
hence ( ((2 (#) h) ") (#) ((f /* (c + h)) - (f /* (c - h))) is convergent & lim (((2 (#) h) ") (#) ((f /* (c + h)) - (f /* (c - h)))) = diff (f,x0) ) by A12, A11, A10, FUNCT_2:63; ::_thesis: verum
end;