let p, g be Real; :: thesis:
let f be one-to-one PartFunc of REAL ,REAL ; :: thesis:
set l = ].p,g.[;
assume that
A0: ].p,g.[ c= dom f and
A1: f is_differentiable_on ].p,g.[ and
A2: ( 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:
A3: rng (f | ].p,g.[) is open by A1, A2, Th41, A0;
set f1 = f | ].p,g.[;
thus f | ].p,g.[ is one-to-one ; :: thesis:
A4: ( dom ((f | ].p,g.[) " ) = rng (f | ].p,g.[) & rng ((f | ].p,g.[) " ) = dom (f | ].p,g.[) ) by FUNCT_1:55;
A5: 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:
assume A6: y0 in dom ((f | ].p,g.[) " ) ; :: thesis:
then consider x0 being Real such that
A7: ( x0 in dom (f | ].p,g.[) & y0 = (f | ].p,g.[) . x0 ) by A4, PARTFUN1:26;
A8: ex N being Neighbourhood of y0 st N c= dom ((f | ].p,g.[) " ) by A3, A4, A6, RCOMP_1:39;
for h being convergent_to_0 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

let h be convergent_to_0 Real_Sequence; :: thesis:
let c be V8() Real_Sequence; :: thesis:
assume A9: ( rng c = {y0} & rng (h + c) c= dom ((f | ].p,g.[) " ) ) ; :: thesis:
reconsider a = NAT --> (((f | ].p,g.[) " ) . y0) as Real_Sequence by FUNCOP_1:57;
reconsider a = a as V8() Real_Sequence ;
A10: rng a = {(((f | ].p,g.[) " ) . y0)}

proof

thus rng a c= {(((f | ].p,g.[) " ) . y0)} :: according to XBOOLE_0:def 10 :: thesis:

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in rng a ; :: thesis:
then ex n being Element of NAT st x = a . n by FUNCT_2:190;
then x = ((f | ].p,g.[) " ) . y0 by FUNCOP_1:13;
hence x in {(((f | ].p,g.[) " ) . y0)} by TARSKI:def 1; :: thesis:

end;

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in {(((f | ].p,g.[) " ) . y0)} ; :: thesis:
then x = ((f | ].p,g.[) " ) . y0 by TARSKI:def 1;
then a . 0 = x by FUNCOP_1:13;
hence x in rng a by VALUED_0:28; :: thesis:

end;

A11: now

let n be Element of NAT ; :: thesis:
c . n in rng c by VALUED_0:28;
hence c . n = (f | ].p,g.[) . x0 by A7, A9, TARSKI:def 1; :: thesis:

end;

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.[) );
A12: for n being Element of NAT ex r being Real st S₁[n,r]

proof

let n be Element of NAT ; :: thesis:
(h + c) . n in rng (h + c) by VALUED_0:28;
then consider g being Real such that
A13: ( g in dom (f | ].p,g.[) & (h + c) . n = (f | ].p,g.[) . g ) by A4, A9, PARTFUN1:26;
A14: a . n = ((f | ].p,g.[) " ) . ((f | ].p,g.[) . x0) by A7, FUNCOP_1:13
.= x0 by A7, FUNCT_1:56 ;
take r = g - x0; :: thesis:
let r1, r2 be real number ; :: thesis:
assume A15: ( r1 = (h + c) . n & r2 = a . n ) ; :: thesis:
hence r1 = f . (r2 + r) by A13, A14, FUNCT_1:70; :: thesis:
g in (dom f) /\ ].p,g.[ by A13, RELAT_1:90;
hence ( r2 + r in dom f & r2 + r in dom (f | ].p,g.[) ) by A13, A14, A15, XBOOLE_0:def 4; :: thesis:

end;

consider b being Real_Sequence such that
A16: for n being Element of NAT holds S₁[n,b . n] from FUNCT_2:sch 3(A12);
A17: ( h is non-zero & h is convergent & lim h = 0 ) by FDIFF_1:def 1;

now

given n being Element of NAT such that A18: b . n = 0 ; :: thesis:
A19: (h + c) . n = (h . n) + (c . n) by SEQ_1:11
.= (h . n) + ((f | ].p,g.[) . x0) by A11 ;
a . n = ((f | ].p,g.[) " ) . ((f | ].p,g.[) . x0) by A7, FUNCOP_1:13
.= x0 by A7, FUNCT_1:56 ;
then f . ((a . n) + (b . n)) = (f | ].p,g.[) . x0 by A7, A18, FUNCT_1:70;
then (h . n) + ((f | ].p,g.[) . x0) = (f | ].p,g.[) . x0 by A16, A19;
hence contradiction by A17, SEQ_1:7; :: thesis:

end;

then A20: b is non-zero by SEQ_1:7;
A21: ( h + c is convergent & lim (h + c) = y0 ) by A9, Th4;
A22: ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing )

proof

now

per cases by A2;

suppose A23: for x0 being Real st x0 in ].p,g.[ holds
0 < diff f,x0 ; :: thesis:

now

per cases = {} or ].p,g.[ <> {} ) ;

suppose ].p,g.[ = {} ; :: thesis:

then ].p,g.[ /\ (dom f) = {} ;
then ].p,g.[ misses dom f by XBOOLE_0:def 7;
hence ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing ) by RFUNCT_2:54; :: thesis:

end;

suppose ].p,g.[ <> {} ; :: thesis:

then p < g by XXREAL_1:28;
hence ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing ) by A1, A23, A0, ROLLE:9; :: thesis:

end;

hence ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing ) ; :: thesis:

end;

suppose A24: for x0 being Real st x0 in ].p,g.[ holds
diff f,x0 < 0 ; :: thesis:

now

per cases = {} or ].p,g.[ <> {} ) ;

suppose ].p,g.[ = {} ; :: thesis:

then ].p,g.[ /\ (dom f) = {} ;
then ].p,g.[ misses dom f by XBOOLE_0:def 7;
hence ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing ) by RFUNCT_2:54; :: thesis:

end;

suppose ].p,g.[ <> {} ; :: thesis:

then p < g by XXREAL_1:28;
hence ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing ) by A1, A24, A0, ROLLE:10; :: thesis:

end;

hence ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing ) ; :: thesis:

end;

hence ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing ) ; :: thesis:

end;

X: y0 in dom (((f | ].p,g.[) " ) | (rng (f | ].p,g.[))) by A4, A6, RELAT_1:98;
].p,g.[ c= dom f by A1, FDIFF_1:def 7;
then ((f | ].p,g.[) " ) | (f .: ].p,g.[) is continuous by A22, FCONT_3:25;
then ((f | ].p,g.[) " ) | (rng (f | ].p,g.[)) is continuous by RELAT_1:148;
then ((f | ].p,g.[) " ) | (rng (f | ].p,g.[)) is_continuous_in y0 by X, FCONT_1:def 2;
then (f | ].p,g.[) " is_continuous_in y0 by A4, RELAT_1:97;
then A25: ( ((f | ].p,g.[) " ) /* (h + c) is convergent & ((f | ].p,g.[) " ) . y0 = lim (((f | ].p,g.[) " ) /* (h + c)) ) by A9, A21, FCONT_1:def 1;
A27: lim a = a . 0 by SEQ_4:41
.= ((f | ].p,g.[) " ) . y0 by FUNCOP_1:13 ;
A28: (((f | ].p,g.[) " ) /* (h + c)) - a is convergent by A25, SEQ_2:25;
A29: lim ((((f | ].p,g.[) " ) /* (h + c)) - a) = (((f | ].p,g.[) " ) . y0) - (((f | ].p,g.[) " ) . y0) by A25, A27, SEQ_2:26
.= 0 ;
A30: rng (b + a) c= dom f

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in rng (b + a) ; :: thesis:
then consider n being Element of NAT such that
A31: x = (b + a) . n by FUNCT_2:190;
A32: ( (h + c) . n = (h + c) . n & a . n = a . n ) ;
x = (a . n) + (b . n) by A31, SEQ_1:11;
hence x in dom f by A16, A32; :: thesis:

end;

now

let n be Element of NAT ; :: thesis:
( (h + c) . n = (h + c) . n & a . n = a . n ) ;
then A33: (a . n) + (b . n) in dom (f | ].p,g.[) by A16;
thus ((((f | ].p,g.[) " ) /* (h + c)) - a) . n = ((((f | ].p,g.[) " ) /* (h + c)) . n) - (a . n) by RFUNCT_2:6
.= (((f | ].p,g.[) " ) . ((h + c) . n)) - (a . n) by A9, FUNCT_2:185
.= (((f | ].p,g.[) " ) . (f . ((a . n) + (b . n)))) - (a . n) by A16
.= (((f | ].p,g.[) " ) . ((f | ].p,g.[) . ((a . n) + (b . n)))) - (a . n) by A33, FUNCT_1:70
.= ((a . n) + (b . n)) - (a . n) by A33, FUNCT_1:56
.= b . n ; :: thesis:

end;

then ( b is convergent & lim b = 0 ) by A28, A29, FUNCT_2:113;
then reconsider b = b as convergent_to_0 Real_Sequence by A20, FDIFF_1:def 1;
A34: rng a c= dom f

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in rng a ; :: thesis:
then x = ((f | ].p,g.[) " ) . y0 by A10, TARSKI:def 1;
then x = x0 by A7, FUNCT_1:56;
then x in (dom f) /\ ].p,g.[ by A7, RELAT_1:90;
hence x in dom f by XBOOLE_0:def 4; :: thesis:

end;

now

let n be Element of NAT ; :: thesis:
c . n in rng c by VALUED_0:28;
then A35: c . n = y0 by A9, TARSKI:def 1;
A36: ((f | ].p,g.[) " ) . y0 in rng ((f | ].p,g.[) " ) by A6, FUNCT_1:def 5;
thus (f /* a) . n = f . (a . n) by A34, FUNCT_2:185
.= f . (((f | ].p,g.[) " ) . y0) by FUNCOP_1:13
.= (f | ].p,g.[) . (((f | ].p,g.[) " ) . y0) by A4, A36, FUNCT_1:70
.= c . n by A4, A6, A35, FUNCT_1:57 ; :: thesis:

end;

then A37: f /* a = c by FUNCT_2:113;

now

let n be Element of NAT ; :: thesis:
(h + c) . n = f . ((a . n) + (b . n)) by A16;
then (h . n) + (c . n) = f . ((a . n) + (b . n)) by SEQ_1:11;
hence h . n = (f . ((b . n) + (a . n))) - ((f /* a) . n) by A37
.= (f . ((b + a) . n)) - ((f /* a) . n) by SEQ_1:11
.= ((f /* (b + a)) . n) - ((f /* a) . n) by A30, FUNCT_2:185
.= ((f /* (b + a)) - (f /* a)) . n by RFUNCT_2:6 ;
:: thesis:

end;

then A38: h = (f /* (b + a)) - (f /* a) by FUNCT_2:113;
then A39: (f /* (b + a)) - (f /* a) is non-zero by FDIFF_1:def 1;
A40: rng c c= dom ((f | ].p,g.[) " )

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in rng c ; :: thesis:
hence x in dom ((f | ].p,g.[) " ) by A6, A9, TARSKI:def 1; :: thesis:

end;

A41: b " is non-zero by A20, SEQ_1:41;

now

let n be Element of NAT ; :: thesis:
( (h + c) . n = (h + c) . n & a . n = a . n ) ;
then A42: (a . n) + (b . n) in dom (f | ].p,g.[) by A16;
c . n in rng c by VALUED_0:28;
then A43: c . n = y0 by A9, 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_1:12
.= ((h " ) . n) * (((((f | ].p,g.[) " ) /* (h + c)) . n) - ((((f | ].p,g.[) " ) /* c) . n)) by RFUNCT_2:6
.= ((h " ) . n) * ((((f | ].p,g.[) " ) . ((h + c) . n)) - ((((f | ].p,g.[) " ) /* c) . n)) by A9, FUNCT_2:185
.= ((h " ) . n) * ((((f | ].p,g.[) " ) . (f . ((a . n) + (b . n)))) - ((((f | ].p,g.[) " ) /* c) . n)) by A16
.= ((h " ) . n) * ((((f | ].p,g.[) " ) . ((f | ].p,g.[) . ((a . n) + (b . n)))) - ((((f | ].p,g.[) " ) /* c) . n)) by A42, FUNCT_1:70
.= ((h " ) . n) * (((a . n) + (b . n)) - ((((f | ].p,g.[) " ) /* c) . n)) by A42, FUNCT_1:56
.= ((h " ) . n) * (((a . n) + (b . n)) - (((f | ].p,g.[) " ) . (c . n))) by A40, FUNCT_2:185
.= ((h " ) . n) * (((a . n) + (b . n)) - (a . n)) by A43, FUNCOP_1:13
.= ((h " ) (#) ((b " ) " )) . n by SEQ_1:12
.= (((b " ) (#) ((f /* (b + a)) - (f /* a))) " ) . n by A38, SEQ_1:44 ; :: thesis:

end;

then A44: (h " ) (#) ((((f | ].p,g.[) " ) /* (h + c)) - (((f | ].p,g.[) " ) /* c)) = ((b " ) (#) ((f /* (b + a)) - (f /* a))) " by FUNCT_2:113;
A45: (b " ) (#) ((f /* (b + a)) - (f /* a)) is non-zero by A39, A41, SEQ_1:43;
((f | ].p,g.[) " ) . y0 in dom (f | ].p,g.[) by A4, A6, FUNCT_1:def 5;
then ((f | ].p,g.[) " ) . y0 in (dom f) /\ ].p,g.[ by RELAT_1:90;
then A46: ((f | ].p,g.[) " ) . y0 in ].p,g.[ by XBOOLE_0:def 4;
then ( f is_differentiable_in ((f | ].p,g.[) " ) . y0 & diff f,(((f | ].p,g.[) " ) . y0) = diff f,(((f | ].p,g.[) " ) . y0) ) by A1, FDIFF_1:16;
then A47: ( (b " ) (#) ((f /* (b + a)) - (f /* a)) is convergent & lim ((b " ) (#) ((f /* (b + a)) - (f /* a))) = diff f,(((f | ].p,g.[) " ) . y0) ) by A10, A30, Th12;
A48: 0 <> diff f,(((f | ].p,g.[) " ) . y0) by A2, A46;
hence (h " ) (#) ((((f | ].p,g.[) " ) /* (h + c)) - (((f | ].p,g.[) " ) /* c)) is convergent by A44, A45, A47, SEQ_2:35; :: thesis:
thus lim ((h " ) (#) ((((f | ].p,g.[) " ) /* (h + c)) - (((f | ].p,g.[) " ) /* c))) = (diff f,(((f | ].p,g.[) " ) . y0)) " by A44, A45, A47, A48, SEQ_2:36
.= 1 / (diff f,(((f | ].p,g.[) " ) . y0)) by XCMPLX_1:217 ; :: thesis:

end;

hence ( (f | ].p,g.[) " is_differentiable_in y0 & diff ((f | ].p,g.[) " ),y0 = 1 / (diff f,(((f | ].p,g.[) " ) . y0)) ) by A8, Th12; :: thesis:

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 A3, A4, FDIFF_1:16; :: thesis:
let x0 be Real; :: thesis:
assume x0 in dom ((f | ].p,g.[) " ) ; :: thesis:
hence diff ((f | ].p,g.[) " ),x0 = 1 / (diff f,(((f | ].p,g.[) " ) . x0)) by A5; :: thesis: