:: SIN_COS9 semantic presentation

begin


theorem Th1: :: SIN_COS9:1
].(- (PI / 2)),(PI / 2).[ c= dom tan
proof
 ].(- (PI / 2)),(PI / 2).[ /\ (cos " {0}) = {}
proof
assume  ].(- (PI / 2)),(PI / 2).[ /\ (cos " {0}) <> {} ; ::_thesis: contradiction
then consider rr being set  such that
A1: rr in ].(- (PI / 2)),(PI / 2).[ /\ (cos " {0}) by XBOOLE_0:7;
 rr in cos " {0} by A1, XBOOLE_0:def_4;
then A2: cos . rr in {0} by FUNCT_1:def_7;
 rr in ].(- (PI / 2)),(PI / 2).[ by A1, XBOOLE_0:def_4;
then  cos . rr <> 0 by COMPTRIG:11;
hence  contradiction by A2, TARSKI:def_1; ::_thesis: verum
end;
then A3: ].(- (PI / 2)),(PI / 2).[ misses cos " {0} by XBOOLE_0:def_7;
 ].(- (PI / 2)),(PI / 2).[ \ (cos " {0}) c= (dom cos) \ (cos " {0}) by SIN_COS:24, XBOOLE_1:33;
then  ].(- (PI / 2)),(PI / 2).[ c= (dom cos) \ (cos " {0}) by A3, XBOOLE_1:83;
then  ].(- (PI / 2)),(PI / 2).[ c= (dom sin) /\ ((dom cos) \ (cos " {0})) by SIN_COS:24, XBOOLE_1:19;
hence  ].(- (PI / 2)),(PI / 2).[ c= dom tan by RFUNCT_1:def_1; ::_thesis: verum
end;

theorem Th2: :: SIN_COS9:2
].0,PI.[ c= dom cot
proof
 ].0,PI.[ /\ (sin " {0}) = {}
proof
assume  ].0,PI.[ /\ (sin " {0}) <> {} ; ::_thesis: contradiction
then consider rr being set  such that
A1: rr in ].0,PI.[ /\ (sin " {0}) by XBOOLE_0:7;
 rr in sin " {0} by A1, XBOOLE_0:def_4;
then A2: sin . rr in {0} by FUNCT_1:def_7;
 rr in ].0,PI.[ by A1, XBOOLE_0:def_4;
then  sin . rr <> 0 by COMPTRIG:7;
hence  contradiction by A2, TARSKI:def_1; ::_thesis: verum
end;
then A3: ].0,PI.[ misses sin " {0} by XBOOLE_0:def_7;
 ].0,PI.[ \ (sin " {0}) c= (dom sin) \ (sin " {0}) by SIN_COS:24, XBOOLE_1:33;
then  ].0,PI.[ c= (dom sin) \ (sin " {0}) by A3, XBOOLE_1:83;
then  ].0,PI.[ c= (dom cos) /\ ((dom sin) \ (sin " {0})) by SIN_COS:24, XBOOLE_1:19;
hence  ].0,PI.[ c= dom cot by RFUNCT_1:def_1; ::_thesis: verum
end;

Lm1:  tan is_differentiable_on ].(- (PI / 2)),(PI / 2).[
proof
 for x being Real st x in ].(- (PI / 2)),(PI / 2).[ holds
tan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in ].(- (PI / 2)),(PI / 2).[ implies tan is_differentiable_in x )
assume  x in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: tan is_differentiable_in x
then  cos . x <> 0 by COMPTRIG:11;
hence  tan is_differentiable_in x by FDIFF_7:46; ::_thesis: verum
end;
hence  tan is_differentiable_on ].(- (PI / 2)),(PI / 2).[ by Th1, FDIFF_1:9; ::_thesis: verum
end;

Lm2:  cot is_differentiable_on ].0,PI.[
proof
 for x being Real st x in ].0,PI.[ holds
cot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in ].0,PI.[ implies cot is_differentiable_in x )
assume  x in ].0,PI.[ ; ::_thesis: cot is_differentiable_in x
then  sin . x <> 0 by COMPTRIG:7;
hence  cot is_differentiable_in x by FDIFF_7:47; ::_thesis: verum
end;
hence  cot is_differentiable_on ].0,PI.[ by Th2, FDIFF_1:9; ::_thesis: verum
end;

Lm3:  for x being Real st x in ].(- (PI / 2)),(PI / 2).[ holds
diff (tan,x) = 1 / ((cos . x) ^2)
proof
let x be Real; ::_thesis: ( x in ].(- (PI / 2)),(PI / 2).[ implies diff (tan,x) = 1 / ((cos . x) ^2) )
assume  x in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: diff (tan,x) = 1 / ((cos . x) ^2)
then  cos . x <> 0 by COMPTRIG:11;
hence  diff (tan,x) = 1 / ((cos . x) ^2) by FDIFF_7:46; ::_thesis: verum
end;

Lm4:  for x being Real st x in ].0,PI.[ holds
diff (cot,x) = - (1 / ((sin . x) ^2))
proof
let x be Real; ::_thesis: ( x in ].0,PI.[ implies diff (cot,x) = - (1 / ((sin . x) ^2)) )
assume  x in ].0,PI.[ ; ::_thesis: diff (cot,x) = - (1 / ((sin . x) ^2))
then  sin . x <> 0 by COMPTRIG:7;
hence  diff (cot,x) = - (1 / ((sin . x) ^2)) by FDIFF_7:47; ::_thesis: verum
end;

theorem :: SIN_COS9:3
( tan is_differentiable_on ].(- (PI / 2)),(PI / 2).[ & ( for x being Real st x in ].(- (PI / 2)),(PI / 2).[ holds
diff (tan,x) = 1 / ((cos . x) ^2) ) ) by Lm1, Lm3;

theorem :: SIN_COS9:4
( cot is_differentiable_on ].0,PI.[ & ( for x being Real st x in ].0,PI.[ holds
diff (cot,x) = - (1 / ((sin . x) ^2)) ) ) by Lm2, Lm4;

theorem :: SIN_COS9:5
tan | ].(- (PI / 2)),(PI / 2).[ is continuous by Lm1, FDIFF_1:25;

theorem :: SIN_COS9:6
cot | ].0,PI.[ is continuous by Lm2, FDIFF_1:25;

theorem Th7: :: SIN_COS9:7
tan | ].(- (PI / 2)),(PI / 2).[ is increasing
proof
A1: for x being Real st x in ].(- (PI / 2)),(PI / 2).[ holds
diff (tan,x) > 0
proof
let x be Real; ::_thesis: ( x in ].(- (PI / 2)),(PI / 2).[ implies diff (tan,x) > 0 )
assume A2: x in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: diff (tan,x) > 0
then  0 < cos . x by COMPTRIG:11;
then  (cos . x) ^2 > 0 by SQUARE_1:12;
then  1 / ((cos . x) ^2) > 0 / ((cos . x) ^2) by XREAL_1:74;
hence  diff (tan,x) > 0 by A2, Lm3; ::_thesis: verum
end;
 ].(- (PI / 2)),(PI / 2).[ c= dom tan by Lm1, FDIFF_1:def_6;
hence  tan | ].(- (PI / 2)),(PI / 2).[ is increasing by A1, Lm1, ROLLE:9; ::_thesis: verum
end;

theorem Th8: :: SIN_COS9:8
cot | ].0,PI.[ is decreasing
proof
A1: for x being Real st x in ].0,PI.[ holds
diff (cot,x) < 0
proof
let x be Real; ::_thesis: ( x in ].0,PI.[ implies diff (cot,x) < 0 )
assume A2: x in ].0,PI.[ ; ::_thesis: diff (cot,x) < 0
then  0 < sin . x by COMPTRIG:7;
then  (sin . x) ^2 > 0 by SQUARE_1:12;
then  1 / ((sin . x) ^2) > 0 / ((sin . x) ^2) by XREAL_1:74;
then  - (1 / ((sin . x) ^2)) < - 0 by XREAL_1:24;
hence  diff (cot,x) < 0 by A2, Lm4; ::_thesis: verum
end;
 ].0,PI.[ c= dom cot by Lm2, FDIFF_1:def_6;
hence  cot | ].0,PI.[ is decreasing by A1, Lm2, ROLLE:10; ::_thesis: verum
end;

theorem :: SIN_COS9:9
tan | ].(- (PI / 2)),(PI / 2).[ is one-to-one by Th7, FCONT_3:8;

theorem :: SIN_COS9:10
cot | ].0,PI.[ is one-to-one by Th8, FCONT_3:8;

registration
clusterK77(tan,].(- (PI / 2)),(PI / 2).[) ->  one-to-one ;
coherence
 tan | ].(- (PI / 2)),(PI / 2).[ is one-to-one by Th7, FCONT_3:8;
clusterK77(cot,].0,PI.[) ->  one-to-one ;
coherence
 cot | ].0,PI.[ is one-to-one by Th8, FCONT_3:8;
end;

definition
func arctan  ->  PartFunc of REAL,REAL equals :: SIN_COS9:def 1
(tan | ].(- (PI / 2)),(PI / 2).[) " ;
coherence
 (tan | ].(- (PI / 2)),(PI / 2).[) " is PartFunc of REAL,REAL ;
func arccot  ->  PartFunc of REAL,REAL equals :: SIN_COS9:def 2
(cot | ].0,PI.[) " ;
coherence
 (cot | ].0,PI.[) " is PartFunc of REAL,REAL ;
end;

:: deftheorem  defines arctan SIN_COS9:def_1_:_
 arctan = (tan | ].(- (PI / 2)),(PI / 2).[) " ;

:: deftheorem  defines arccot SIN_COS9:def_2_:_
 arccot = (cot | ].0,PI.[) " ;

definition
let r be Real;
func arctan r ->  set  equals :: SIN_COS9:def 3
arctan . r;
coherence
 arctan . r is set ;
func arccot r ->  set  equals :: SIN_COS9:def 4
arccot . r;
coherence
 arccot . r is set ;
end;

:: deftheorem  defines arctan SIN_COS9:def_3_:_
 for r being Real holds arctan r = arctan . r;

:: deftheorem  defines arccot SIN_COS9:def_4_:_
 for r being Real holds arccot r = arccot . r;

definition
let r be Real;
:: original: arctan
redefine func arctan r ->  Real;
coherence
 arctan r is Real ;
:: original: arccot
redefine func arccot r ->  Real;
coherence
 arccot r is Real ;
end;

Lm5:  arctan " = tan | ].(- (PI / 2)),(PI / 2).[
by FUNCT_1:43;

Lm6:  arccot " = cot | ].0,PI.[
by FUNCT_1:43;

theorem Th11: :: SIN_COS9:11
rng arctan = ].(- (PI / 2)),(PI / 2).[
proof
 dom (tan | ].(- (PI / 2)),(PI / 2).[) = ].(- (PI / 2)),(PI / 2).[ by Th1, RELAT_1:62;
hence  rng arctan = ].(- (PI / 2)),(PI / 2).[ by FUNCT_1:33; ::_thesis: verum
end;

theorem Th12: :: SIN_COS9:12
rng arccot = ].0,PI.[
proof
 dom (cot | ].0,PI.[) = ].0,PI.[ by Th2, RELAT_1:62;
hence  rng arccot = ].0,PI.[ by FUNCT_1:33; ::_thesis: verum
end;

registration
cluster arctan  ->  one-to-one ;
coherence
 arctan is one-to-one ;
cluster arccot  ->  one-to-one ;
coherence
 arccot is one-to-one ;
end;

Lm7:  - (PI / 4) in ].(- (PI / 2)),(PI / 2).[
proof
 PI in ].0,4.[ by SIN_COS:def_28;
then  PI > 0 by XXREAL_1:4;
then  PI / (- 4) > PI / (- 2) by XREAL_1:99;
then  - (PI / 4) in  {  s where s is Real : ( - (PI / 2) < s & s < PI / 2 )  }  ;
hence  - (PI / 4) in ].(- (PI / 2)),(PI / 2).[ by RCOMP_1:def_2; ::_thesis: verum
end;

Lm8:  PI / 4 in ].(- (PI / 2)),(PI / 2).[
proof
 PI in ].0,4.[ by SIN_COS:def_28;
then  PI > 0 by XXREAL_1:4;
then  PI / 4 < PI / 2 by XREAL_1:76;
then  PI / 4 in  {  s where s is Real : ( - (PI / 2) < s & s < PI / 2 )  }  ;
hence  PI / 4 in ].(- (PI / 2)),(PI / 2).[ by RCOMP_1:def_2; ::_thesis: verum
end;

Lm9:  PI / 4 in ].0,PI.[
proof
 PI in ].0,4.[ by SIN_COS:def_28;
then A1: PI > 0 by XXREAL_1:4;
then A2: PI / 4 < PI / 1 by XREAL_1:76;
 PI / 4 > 0 / 4 by A1, XREAL_1:74;
hence  PI / 4 in ].0,PI.[ by A2, XXREAL_1:4; ::_thesis: verum
end;

Lm10:  (3 / 4) * PI in ].0,PI.[
proof
 PI in ].0,4.[ by SIN_COS:def_28;
then A1: PI > 0 by XXREAL_1:4;
then A2: (3 / 4) * PI < PI by XREAL_1:157;
 (3 / 4) * PI > (3 / 4) * 0 by A1, XREAL_1:68;
hence  (3 / 4) * PI in ].0,PI.[ by A2, XXREAL_1:4; ::_thesis: verum
end;

Lm11:  dom (tan | [.(- (PI / 4)),(PI / 4).]) = [.(- (PI / 4)),(PI / 4).]
proof
 [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12;
hence  dom (tan | [.(- (PI / 4)),(PI / 4).]) = [.(- (PI / 4)),(PI / 4).] by Th1, RELAT_1:62, XBOOLE_1:1; ::_thesis: verum
end;

Lm12:  dom (cot | [.(PI / 4),((3 / 4) * PI).]) = [.(PI / 4),((3 / 4) * PI).]
proof
 [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12;
hence  dom (cot | [.(PI / 4),((3 / 4) * PI).]) = [.(PI / 4),((3 / 4) * PI).] by Th2, RELAT_1:62, XBOOLE_1:1; ::_thesis: verum
end;

theorem Th13: :: SIN_COS9:13
for x being real number st x in ].(- (PI / 2)),(PI / 2).[ holds
tan . x = tan x
proof
let x be real number ; ::_thesis: ( x in ].(- (PI / 2)),(PI / 2).[ implies tan . x = tan x )
assume  x in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: tan . x = tan x
then tan . x = (sin x) / (cos x) by Th1, RFUNCT_1:def_1
.= tan x by SIN_COS4:def_1 ;
hence  tan . x = tan x ; ::_thesis: verum
end;

theorem Th14: :: SIN_COS9:14
for x being real number st x in ].0,PI.[ holds
cot . x = cot x
proof
let x be real number ; ::_thesis: ( x in ].0,PI.[ implies cot . x = cot x )
assume  x in ].0,PI.[ ; ::_thesis: cot . x = cot x
then cot . x = (cos x) / (sin x) by Th2, RFUNCT_1:def_1
.= cot x by SIN_COS4:def_2 ;
hence  cot . x = cot x ; ::_thesis: verum
end;

theorem :: SIN_COS9:15
for x being Real st cos . x <> 0 holds
tan . x = tan x
proof
let x be Real; ::_thesis: ( cos . x <> 0 implies tan . x = tan x )
assume A1: cos . x <> 0 ; ::_thesis: tan . x = tan x
 not x in cos " {0}
proof
assume  x in cos " {0} ; ::_thesis: contradiction
then  cos . x in {0} by FUNCT_1:def_7;
hence  contradiction by A1, TARSKI:def_1; ::_thesis: verum
end;
then  x in (dom cos) \ (cos " {0}) by SIN_COS:24, XBOOLE_0:def_5;
then  x in (dom sin) /\ ((dom cos) \ (cos " {0})) by SIN_COS:24, XBOOLE_0:def_4;
then  x in dom (sin / cos) by RFUNCT_1:def_1;
then tan . x = (sin x) / (cos x) by RFUNCT_1:def_1
.= tan x by SIN_COS4:def_1 ;
hence  tan . x = tan x ; ::_thesis: verum
end;

theorem :: SIN_COS9:16
for x being Real st sin . x <> 0 holds
cot . x = cot x
proof
let x be Real; ::_thesis: ( sin . x <> 0 implies cot . x = cot x )
assume A1: sin . x <> 0 ; ::_thesis: cot . x = cot x
 not x in sin " {0}
proof
assume  x in sin " {0} ; ::_thesis: contradiction
then  sin . x in {0} by FUNCT_1:def_7;
hence  contradiction by A1, TARSKI:def_1; ::_thesis: verum
end;
then  x in (dom sin) \ (sin " {0}) by SIN_COS:24, XBOOLE_0:def_5;
then  x in (dom cos) /\ ((dom sin) \ (sin " {0})) by SIN_COS:24, XBOOLE_0:def_4;
then  x in dom (cos / sin) by RFUNCT_1:def_1;
then cot . x = (cos x) / (sin x) by RFUNCT_1:def_1
.= cot x by SIN_COS4:def_2 ;
hence  cot . x = cot x ; ::_thesis: verum
end;

theorem Th17: :: SIN_COS9:17
( tan . (- (PI / 4)) = - 1 & tan (- (PI / 4)) = - 1 )
proof
 cos . (PI / 4) <> 0 by Lm8, COMPTRIG:11;
then A1: (sin . (PI / 4)) / (cos . (PI / 4)) = 1 by SIN_COS:73, XCMPLX_1:60;
tan . (- (PI / 4)) = (sin . (- (PI / 4))) / (cos . (- (PI / 4))) by Lm7, Th1, RFUNCT_1:def_1
.= (- (sin . (PI / 4))) / (cos . (- (PI / 4))) by SIN_COS:30
.= (- (sin . (PI / 4))) / (cos . (PI / 4)) by SIN_COS:30
.= - 1 by A1 ;
hence  ( tan . (- (PI / 4)) = - 1 & tan (- (PI / 4)) = - 1 ) by Lm7, Th13; ::_thesis: verum
end;

theorem Th18: :: SIN_COS9:18
( cot . (PI / 4) = 1 & cot (PI / 4) = 1 & cot . ((3 / 4) * PI) = - 1 & cot ((3 / 4) * PI) = - 1 )
proof
A1: sin . (PI / 4) <> 0 by Lm9, COMPTRIG:7;
A2: cot . ((3 / 4) * PI) = (cos . ((3 / 4) * PI)) * ((sin . ((3 / 4) * PI)) ") by Lm10, Th2, RFUNCT_1:def_1
.= (- (sin . (PI / 4))) / (sin . ((PI / 2) + (PI / 4))) by SIN_COS:78
.= (- (sin . (PI / 4))) / (cos . (PI / 4)) by SIN_COS:78
.= - ((sin . (PI / 4)) / (cos . (PI / 4)))
.= - 1 by A1, SIN_COS:73, XCMPLX_1:60 ;
cot . (PI / 4) = (cos . (PI / 4)) / (sin . (PI / 4)) by Lm9, Th2, RFUNCT_1:def_1
.= 1 by A1, SIN_COS:73, XCMPLX_1:60 ;
hence  ( cot . (PI / 4) = 1 & cot (PI / 4) = 1 & cot . ((3 / 4) * PI) = - 1 & cot ((3 / 4) * PI) = - 1 ) by A2, Lm9, Lm10, Th14; ::_thesis: verum
end;

theorem Th19: :: SIN_COS9:19
for x being set st x in [.(- (PI / 4)),(PI / 4).] holds
tan . x in [.(- 1),1.]
proof
let x be set ; ::_thesis: ( x in [.(- (PI / 4)),(PI / 4).] implies tan . x in [.(- 1),1.] )
assume  x in [.(- (PI / 4)),(PI / 4).] ; ::_thesis: tan . x in [.(- 1),1.]
then  x in ].(- (PI / 4)),(PI / 4).[ \/ {(- (PI / 4)),(PI / 4)} by XXREAL_1:128;
then A1: ( x in ].(- (PI / 4)),(PI / 4).[ or x in {(- (PI / 4)),(PI / 4)} ) by XBOOLE_0:def_3;
percases ( x in ].(- (PI / 4)),(PI / 4).[ or x = - (PI / 4) or x = PI / 4 ) by A1, TARSKI:def_2;
supposeA2: x in ].(- (PI / 4)),(PI / 4).[ ; ::_thesis: tan . x in [.(- 1),1.]
then  x in  {  s where s is Real : ( - (PI / 4) < s & s < PI / 4 )  }  by RCOMP_1:def_2;
then A3: ex s being Real st
( s = x & - (PI / 4) < s & s < PI / 4 ) ;
A4: ].(- (PI / 4)),(PI / 4).[ c= [.(- (PI / 4)),(PI / 4).] by XXREAL_1:25;
 - (PI / 4) in  {  s where s is Real : ( - (PI / 4) <= s & s <= PI / 4 )  }  ;
then A5: - (PI / 4) in [.(- (PI / 4)),(PI / 4).] by RCOMP_1:def_1;
A6: [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12;
then A7: tan | [.(- (PI / 4)),(PI / 4).] is increasing by Th7, RFUNCT_2:28;
A8: [.(- (PI / 4)),(PI / 4).] /\ (dom tan) = [.(- (PI / 4)),(PI / 4).] by A6, Th1, XBOOLE_1:1, XBOOLE_1:28;
 PI / 4 in  {  s where s is Real : ( - (PI / 4) <= s & s <= PI / 4 )  }  ;
then  PI / 4 in [.(- (PI / 4)),(PI / 4).] /\ (dom tan) by A8, RCOMP_1:def_1;
then  tan . x < tan . (PI / 4) by A2, A7, A8, A4, A3, RFUNCT_2:20;
then A9: tan . x < 1 by SIN_COS:def_28;
 x in  {  s where s is Real : ( - (PI / 4) < s & s < PI / 4 )  }  by A2, RCOMP_1:def_2;
then  ex s being Real st
( s = x & - (PI / 4) < s & s < PI / 4 ) ;
then  - 1 < tan . x by A2, A7, A5, A8, A4, Th17, RFUNCT_2:20;
then  tan . x in  {  s where s is Real : ( - 1 < s & s < 1 )  }  by A9;
then A10: tan . x in ].(- 1),1.[ by RCOMP_1:def_2;
 ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
hence  tan . x in [.(- 1),1.] by A10; ::_thesis: verum
end;
suppose x = - (PI / 4) ; ::_thesis: tan . x in [.(- 1),1.]
then  tan . x in  {  s where s is Real : ( - 1 <= s & s <= 1 )  }  by Th17;
hence  tan . x in [.(- 1),1.] by RCOMP_1:def_1; ::_thesis: verum
end;
suppose x = PI / 4 ; ::_thesis: tan . x in [.(- 1),1.]
then  tan . x = 1 by SIN_COS:def_28;
then  tan . x in  {  s where s is Real : ( - 1 <= s & s <= 1 )  }  ;
hence  tan . x in [.(- 1),1.] by RCOMP_1:def_1; ::_thesis: verum
end;
end;
end;

theorem Th20: :: SIN_COS9:20
for x being set st x in [.(PI / 4),((3 / 4) * PI).] holds
cot . x in [.(- 1),1.]
proof
let x be set ; ::_thesis: ( x in [.(PI / 4),((3 / 4) * PI).] implies cot . x in [.(- 1),1.] )
 PI in ].0,4.[ by SIN_COS:def_28;
then  PI > 0 by XXREAL_1:4;
then  PI / 4 > 0 / 4 by XREAL_1:74;
then A1: (PI / 4) * 3 > PI / 4 by XREAL_1:155;
assume  x in [.(PI / 4),((3 / 4) * PI).] ; ::_thesis: cot . x in [.(- 1),1.]
then  x in ].(PI / 4),((3 / 4) * PI).[ \/ {(PI / 4),((3 / 4) * PI)} by A1, XXREAL_1:128;
then A2: ( x in ].(PI / 4),((3 / 4) * PI).[ or x in {(PI / 4),((3 / 4) * PI)} ) by XBOOLE_0:def_3;
percases ( x in ].(PI / 4),((3 / 4) * PI).[ or x = PI / 4 or x = (3 / 4) * PI ) by A2, TARSKI:def_2;
supposeA3: x in ].(PI / 4),((3 / 4) * PI).[ ; ::_thesis: cot . x in [.(- 1),1.]
then  x in  {  s where s is Real : ( PI / 4 < s & s < (3 / 4) * PI )  }  by RCOMP_1:def_2;
then A4: ex s being Real st
( s = x & PI / 4 < s & s < (3 / 4) * PI ) ;
A5: [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12;
then A6: cot | [.(PI / 4),((3 / 4) * PI).] is decreasing by Th8, RFUNCT_2:29;
 x in  {  s where s is Real : ( PI / 4 < s & s < (3 / 4) * PI )  }  by A3, RCOMP_1:def_2;
then A7: ex s being Real st
( s = x & PI / 4 < s & s < (3 / 4) * PI ) ;
A8: ].(PI / 4),((3 / 4) * PI).[ c= [.(PI / 4),((3 / 4) * PI).] by XXREAL_1:25;
A9: [.(PI / 4),((3 / 4) * PI).] /\ (dom cot) = [.(PI / 4),((3 / 4) * PI).] by A5, Th2, XBOOLE_1:1, XBOOLE_1:28;
 (3 / 4) * PI in  {  s where s is Real : ( PI / 4 <= s & s <= (3 / 4) * PI )  }  by A1;
then  (3 / 4) * PI in [.(PI / 4),((3 / 4) * PI).] /\ (dom cot) by A9, RCOMP_1:def_1;
then A10: - 1 < cot . x by A3, A6, A9, A8, A7, Th18, RFUNCT_2:21;
 PI / 4 in  {  s where s is Real : ( PI / 4 <= s & s <= (3 / 4) * PI )  }  by A1;
then  PI / 4 in [.(PI / 4),((3 / 4) * PI).] /\ (dom cot) by A9, RCOMP_1:def_1;
then  cot . x < 1 by A3, A6, A9, A8, A4, Th18, RFUNCT_2:21;
then  cot . x in  {  s where s is Real : ( - 1 < s & s < 1 )  }  by A10;
then A11: cot . x in ].(- 1),1.[ by RCOMP_1:def_2;
 ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
hence  cot . x in [.(- 1),1.] by A11; ::_thesis: verum
end;
suppose x = PI / 4 ; ::_thesis: cot . x in [.(- 1),1.]
then  cot . x in  {  s where s is Real : ( - 1 <= s & s <= 1 )  }  by Th18;
hence  cot . x in [.(- 1),1.] by RCOMP_1:def_1; ::_thesis: verum
end;
suppose x = (3 / 4) * PI ; ::_thesis: cot . x in [.(- 1),1.]
then  cot . x in  {  s where s is Real : ( - 1 <= s & s <= 1 )  }  by Th18;
hence  cot . x in [.(- 1),1.] by RCOMP_1:def_1; ::_thesis: verum
end;
end;
end;

theorem Th21: :: SIN_COS9:21
rng (tan | [.(- (PI / 4)),(PI / 4).]) = [.(- 1),1.]
proof
now__::_thesis:_for_y_being_set_holds_
(_(_y_in_[.(-_1),1.]_implies_ex_x_being_set_st_
(_x_in_dom_(tan_|_[.(-_(PI_/_4)),(PI_/_4).])_&_y_=_(tan_|_[.(-_(PI_/_4)),(PI_/_4).])_._x_)_)_&_(_ex_x_being_set_st_
(_x_in_dom_(tan_|_[.(-_(PI_/_4)),(PI_/_4).])_&_y_=_(tan_|_[.(-_(PI_/_4)),(PI_/_4).])_._x_)_implies_y_in_[.(-_1),1.]_)_)
let y be set ; ::_thesis: ( ( y in [.(- 1),1.] implies ex x being set st
( x in dom (tan | [.(- (PI / 4)),(PI / 4).]) & y = (tan | [.(- (PI / 4)),(PI / 4).]) . x ) ) & ( ex x being set st
( x in dom (tan | [.(- (PI / 4)),(PI / 4).]) & y = (tan | [.(- (PI / 4)),(PI / 4).]) . x ) implies y in [.(- 1),1.] ) )
thus  ( y in [.(- 1),1.] implies ex x being set st
( x in dom (tan | [.(- (PI / 4)),(PI / 4).]) & y = (tan | [.(- (PI / 4)),(PI / 4).]) . x ) ) ::_thesis: ( ex x being set st
( x in dom (tan | [.(- (PI / 4)),(PI / 4).]) & y = (tan | [.(- (PI / 4)),(PI / 4).]) . x ) implies y in [.(- 1),1.] )
proof
assume A1: y in [.(- 1),1.] ; ::_thesis: ex x being set st
( x in dom (tan | [.(- (PI / 4)),(PI / 4).]) & y = (tan | [.(- (PI / 4)),(PI / 4).]) . x )
then reconsider y1 = y as Real ;
 y1 in [.(tan . (- (PI / 4))),(tan . (PI / 4)).] by A1, Th17, SIN_COS:def_28;
then A2: y1 in [.(tan . (- (PI / 4))),(tan . (PI / 4)).] \/ [.(tan . (PI / 4)),(tan . (- (PI / 4))).] by XBOOLE_0:def_3;
A3: [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12;
 tan | ].(- (PI / 2)),(PI / 2).[ is continuous by Lm1, FDIFF_1:25;
then  tan | [.(- (PI / 4)),(PI / 4).] is continuous by A3, FCONT_1:16;
then consider x being Real such that
A4: x in [.(- (PI / 4)),(PI / 4).] and
A5: y1 = tan . x by A3, A2, Th1, FCONT_2:15, XBOOLE_1:1;
take  x ; ::_thesis: ( x in dom (tan | [.(- (PI / 4)),(PI / 4).]) & y = (tan | [.(- (PI / 4)),(PI / 4).]) . x )
thus  ( x in dom (tan | [.(- (PI / 4)),(PI / 4).]) & y = (tan | [.(- (PI / 4)),(PI / 4).]) . x ) by A4, A5, Lm11, FUNCT_1:49; ::_thesis: verum
end;
thus  ( ex x being set st
( x in dom (tan | [.(- (PI / 4)),(PI / 4).]) & y = (tan | [.(- (PI / 4)),(PI / 4).]) . x ) implies y in [.(- 1),1.] ) ::_thesis: verum
proof
given x being set  such that A6: x in dom (tan | [.(- (PI / 4)),(PI / 4).]) and
A7: y = (tan | [.(- (PI / 4)),(PI / 4).]) . x ; ::_thesis: y in [.(- 1),1.]
reconsider x1 = x as Real by A6;
 y = tan . x1 by A6, A7, Lm11, FUNCT_1:49;
hence  y in [.(- 1),1.] by A6, Lm11, Th19; ::_thesis: verum
end;
end;
hence  rng (tan | [.(- (PI / 4)),(PI / 4).]) = [.(- 1),1.] by FUNCT_1:def_3; ::_thesis: verum
end;

theorem Th22: :: SIN_COS9:22
rng (cot | [.(PI / 4),((3 / 4) * PI).]) = [.(- 1),1.]
proof
now__::_thesis:_for_y_being_set_holds_
(_(_y_in_[.(-_1),1.]_implies_ex_x_being_set_st_
(_x_in_dom_(cot_|_[.(PI_/_4),((3_/_4)_*_PI).])_&_y_=_(cot_|_[.(PI_/_4),((3_/_4)_*_PI).])_._x_)_)_&_(_ex_x_being_set_st_
(_x_in_dom_(cot_|_[.(PI_/_4),((3_/_4)_*_PI).])_&_y_=_(cot_|_[.(PI_/_4),((3_/_4)_*_PI).])_._x_)_implies_y_in_[.(-_1),1.]_)_)
let y be set ; ::_thesis: ( ( y in [.(- 1),1.] implies ex x being set st
( x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) & y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x ) ) & ( ex x being set st
( x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) & y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x ) implies y in [.(- 1),1.] ) )
thus  ( y in [.(- 1),1.] implies ex x being set st
( x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) & y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x ) ) ::_thesis: ( ex x being set st
( x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) & y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x ) implies y in [.(- 1),1.] )
proof
 PI in ].0,4.[ by SIN_COS:def_28;
then  PI > 0 by XXREAL_1:4;
then  PI / 4 > 0 / 4 by XREAL_1:74;
then A1: (PI / 4) * 3 > PI / 4 by XREAL_1:155;
assume A2: y in [.(- 1),1.] ; ::_thesis: ex x being set st
( x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) & y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x )
then reconsider y1 = y as Real ;
A3: y1 in [.(cot . ((3 / 4) * PI)),(cot . (PI / 4)).] \/ [.(cot . (PI / 4)),(cot . ((3 / 4) * PI)).] by A2, Th18, XBOOLE_0:def_3;
A4: [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12;
 cot | ].0,PI.[ is continuous by Lm2, FDIFF_1:25;
then  cot | [.(PI / 4),((3 / 4) * PI).] is continuous by A4, FCONT_1:16;
then consider x being Real such that
A5: x in [.(PI / 4),((3 / 4) * PI).] and
A6: y1 = cot . x by A1, A4, A3, Th2, FCONT_2:15, XBOOLE_1:1;
take  x ; ::_thesis: ( x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) & y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x )
thus  ( x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) & y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x ) by A5, A6, Lm12, FUNCT_1:49; ::_thesis: verum
end;
thus  ( ex x being set st
( x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) & y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x ) implies y in [.(- 1),1.] ) ::_thesis: verum
proof
given x being set  such that A7: x in dom (cot | [.(PI / 4),((3 / 4) * PI).]) and
A8: y = (cot | [.(PI / 4),((3 / 4) * PI).]) . x ; ::_thesis: y in [.(- 1),1.]
reconsider x1 = x as Real by A7;
 y = cot . x1 by A7, A8, Lm12, FUNCT_1:49;
hence  y in [.(- 1),1.] by A7, Lm12, Th20; ::_thesis: verum
end;
end;
hence  rng (cot | [.(PI / 4),((3 / 4) * PI).]) = [.(- 1),1.] by FUNCT_1:def_3; ::_thesis: verum
end;

theorem Th23: :: SIN_COS9:23
[.(- 1),1.] c= dom arctan
proof
A1: [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12;
 rng (tan | [.(- (PI / 4)),(PI / 4).]) c= rng (tan | ].(- (PI / 2)),(PI / 2).[)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (tan | [.(- (PI / 4)),(PI / 4).]) or y in rng (tan | ].(- (PI / 2)),(PI / 2).[) )
assume  y in rng (tan | [.(- (PI / 4)),(PI / 4).]) ; ::_thesis: y in rng (tan | ].(- (PI / 2)),(PI / 2).[)
then  y in tan .: [.(- (PI / 4)),(PI / 4).] by RELAT_1:115;
then  ex x being set st
( x in dom tan & x in [.(- (PI / 4)),(PI / 4).] & y = tan . x ) by FUNCT_1:def_6;
then  y in tan .: ].(- (PI / 2)),(PI / 2).[ by A1, FUNCT_1:def_6;
hence  y in rng (tan | ].(- (PI / 2)),(PI / 2).[) by RELAT_1:115; ::_thesis: verum
end;
hence  [.(- 1),1.] c= dom arctan by Th21, FUNCT_1:33; ::_thesis: verum
end;

theorem Th24: :: SIN_COS9:24
[.(- 1),1.] c= dom arccot
proof
A1: [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12;
 rng (cot | [.(PI / 4),((3 / 4) * PI).]) c= rng (cot | ].0,PI.[)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (cot | [.(PI / 4),((3 / 4) * PI).]) or y in rng (cot | ].0,PI.[) )
assume  y in rng (cot | [.(PI / 4),((3 / 4) * PI).]) ; ::_thesis: y in rng (cot | ].0,PI.[)
then  y in cot .: [.(PI / 4),((3 / 4) * PI).] by RELAT_1:115;
then  ex x being set st
( x in dom cot & x in [.(PI / 4),((3 / 4) * PI).] & y = cot . x ) by FUNCT_1:def_6;
then  y in cot .: ].0,PI.[ by A1, FUNCT_1:def_6;
hence  y in rng (cot | ].0,PI.[) by RELAT_1:115; ::_thesis: verum
end;
hence  [.(- 1),1.] c= dom arccot by Th22, FUNCT_1:33; ::_thesis: verum
end;

Lm13:  (tan | [.(- (PI / 4)),(PI / 4).]) | [.(- (PI / 4)),(PI / 4).] is increasing
proof
 [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12;
then  tan | [.(- (PI / 4)),(PI / 4).] is increasing by Th7, RFUNCT_2:28;
hence  (tan | [.(- (PI / 4)),(PI / 4).]) | [.(- (PI / 4)),(PI / 4).] is increasing ; ::_thesis: verum
end;

Lm14:  (cot | [.(PI / 4),((3 / 4) * PI).]) | [.(PI / 4),((3 / 4) * PI).] is decreasing
proof
 [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12;
then  cot | [.(PI / 4),((3 / 4) * PI).] is decreasing by Th8, RFUNCT_2:29;
hence  (cot | [.(PI / 4),((3 / 4) * PI).]) | [.(PI / 4),((3 / 4) * PI).] is decreasing ; ::_thesis: verum
end;

Lm15:  tan | [.(- (PI / 4)),(PI / 4).] is one-to-one
proof
 [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12;
then  (tan | ].(- (PI / 2)),(PI / 2).[) | [.(- (PI / 4)),(PI / 4).] = tan | [.(- (PI / 4)),(PI / 4).] by RELAT_1:74;
hence  tan | [.(- (PI / 4)),(PI / 4).] is one-to-one ; ::_thesis: verum
end;

Lm16:  cot | [.(PI / 4),((3 / 4) * PI).] is one-to-one
proof
 [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12;
then  (cot | ].0,PI.[) | [.(PI / 4),((3 / 4) * PI).] = cot | [.(PI / 4),((3 / 4) * PI).] by RELAT_1:74;
hence  cot | [.(PI / 4),((3 / 4) * PI).] is one-to-one ; ::_thesis: verum
end;

registration
clusterK77(tan,[.(- (PI / 4)),(PI / 4).]) ->  one-to-one ;
coherence
 tan | [.(- (PI / 4)),(PI / 4).] is one-to-one by Lm15;
clusterK77(cot,[.(PI / 4),((3 / 4) * PI).]) ->  one-to-one ;
coherence
 cot | [.(PI / 4),((3 / 4) * PI).] is one-to-one by Lm16;
end;

theorem Th25: :: SIN_COS9:25
arctan | [.(- 1),1.] = (tan | [.(- (PI / 4)),(PI / 4).]) "
proof
set h = tan | ].(- (PI / 2)),(PI / 2).[;
A1: [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12;
then (tan | [.(- (PI / 4)),(PI / 4).]) " = ((tan | ].(- (PI / 2)),(PI / 2).[) | [.(- (PI / 4)),(PI / 4).]) " by RELAT_1:74
.= ((tan | ].(- (PI / 2)),(PI / 2).[) ") | ((tan | ].(- (PI / 2)),(PI / 2).[) .: [.(- (PI / 4)),(PI / 4).]) by RFUNCT_2:17
.= ((tan | ].(- (PI / 2)),(PI / 2).[) ") | (rng ((tan | ].(- (PI / 2)),(PI / 2).[) | [.(- (PI / 4)),(PI / 4).])) by RELAT_1:115
.= ((tan | ].(- (PI / 2)),(PI / 2).[) ") | [.(- 1),1.] by A1, Th21, RELAT_1:74 ;
hence  arctan | [.(- 1),1.] = (tan | [.(- (PI / 4)),(PI / 4).]) " ; ::_thesis: verum
end;

theorem Th26: :: SIN_COS9:26
arccot | [.(- 1),1.] = (cot | [.(PI / 4),((3 / 4) * PI).]) "
proof
set h = cot | ].0,PI.[;
A1: [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12;
then (cot | [.(PI / 4),((3 / 4) * PI).]) " = ((cot | ].0,PI.[) | [.(PI / 4),((3 / 4) * PI).]) " by RELAT_1:74
.= ((cot | ].0,PI.[) ") | ((cot | ].0,PI.[) .: [.(PI / 4),((3 / 4) * PI).]) by RFUNCT_2:17
.= ((cot | ].0,PI.[) ") | (rng ((cot | ].0,PI.[) | [.(PI / 4),((3 / 4) * PI).])) by RELAT_1:115
.= ((cot | ].0,PI.[) ") | [.(- 1),1.] by A1, Th22, RELAT_1:74 ;
hence  arccot | [.(- 1),1.] = (cot | [.(PI / 4),((3 / 4) * PI).]) " ; ::_thesis: verum
end;

theorem :: SIN_COS9:27
(tan | [.(- (PI / 4)),(PI / 4).]) * (arctan | [.(- 1),1.]) = id [.(- 1),1.] by Th21, Th25, FUNCT_1:39;

theorem :: SIN_COS9:28
(cot | [.(PI / 4),((3 / 4) * PI).]) * (arccot | [.(- 1),1.]) = id [.(- 1),1.] by Th22, Th26, FUNCT_1:39;

theorem :: SIN_COS9:29
(tan | [.(- (PI / 4)),(PI / 4).]) * (arctan | [.(- 1),1.]) = id [.(- 1),1.] by Th21, Th25, FUNCT_1:39;

theorem :: SIN_COS9:30
(cot | [.(PI / 4),((3 / 4) * PI).]) * (arccot | [.(- 1),1.]) = id [.(- 1),1.] by Th22, Th26, FUNCT_1:39;

theorem Th31: :: SIN_COS9:31
arctan * (tan | ].(- (PI / 2)),(PI / 2).[) = id ].(- (PI / 2)),(PI / 2).[ by Lm5, Th11, FUNCT_1:39;

theorem Th32: :: SIN_COS9:32
arccot * (cot | ].0,PI.[) = id ].0,PI.[ by Lm6, Th12, FUNCT_1:39;

theorem :: SIN_COS9:33
arctan * (tan | ].(- (PI / 2)),(PI / 2).[) = id ].(- (PI / 2)),(PI / 2).[ by Lm5, Th11, FUNCT_1:39;

theorem :: SIN_COS9:34
arccot * (cot | ].0,PI.[) = id ].0,PI.[ by Lm6, Th12, FUNCT_1:39;

theorem Th35: :: SIN_COS9:35
for r being Real st - (PI / 2) < r & r < PI / 2 holds
( arctan (tan . r) = r & arctan (tan r) = r )
proof
let r be Real; ::_thesis: ( - (PI / 2) < r & r < PI / 2 implies ( arctan (tan . r) = r & arctan (tan r) = r ) )
assume that
A1: - (PI / 2) < r and
A2: r < PI / 2 ; ::_thesis: ( arctan (tan . r) = r & arctan (tan r) = r )
A3: dom (tan | ].(- (PI / 2)),(PI / 2).[) = ].(- (PI / 2)),(PI / 2).[ by Th1, RELAT_1:62;
A4: r in ].(- (PI / 2)),(PI / 2).[ by A1, A2, XXREAL_1:4;
then  arctan (tan . r) = arctan . ((tan | ].(- (PI / 2)),(PI / 2).[) . r) by FUNCT_1:49
.= (id ].(- (PI / 2)),(PI / 2).[) . r by A4, A3, Th31, FUNCT_1:13
.= r by A4, FUNCT_1:18 ;
hence  ( arctan (tan . r) = r & arctan (tan r) = r ) by A4, Th13; ::_thesis: verum
end;

theorem Th36: :: SIN_COS9:36
for r being Real st 0 < r & r < PI holds
( arccot (cot . r) = r & arccot (cot r) = r )
proof
let r be Real; ::_thesis: ( 0 < r & r < PI implies ( arccot (cot . r) = r & arccot (cot r) = r ) )
assume that
A1: 0 < r and
A2: r < PI ; ::_thesis: ( arccot (cot . r) = r & arccot (cot r) = r )
A3: dom (cot | ].0,PI.[) = ].0,PI.[ by Th2, RELAT_1:62;
A4: r in ].0,PI.[ by A1, A2, XXREAL_1:4;
then  arccot (cot . r) = arccot . ((cot | ].0,PI.[) . r) by FUNCT_1:49
.= (id ].0,PI.[) . r by A4, A3, Th32, FUNCT_1:13
.= r by A4, FUNCT_1:18 ;
hence  ( arccot (cot . r) = r & arccot (cot r) = r ) by A4, Th14; ::_thesis: verum
end;

theorem Th37: :: SIN_COS9:37
( arctan (- 1) = - (PI / 4) & arctan . (- 1) = - (PI / 4) )
proof
 - (PI / 2) < - (PI / 4) by Lm7, XXREAL_1:4;
then  arctan (- 1) = - (PI / 4) by Th17, Th35;
hence  ( arctan (- 1) = - (PI / 4) & arctan . (- 1) = - (PI / 4) ) ; ::_thesis: verum
end;

theorem Th38: :: SIN_COS9:38
( arccot (- 1) = (3 / 4) * PI & arccot . (- 1) = (3 / 4) * PI )
proof
A1: (3 / 4) * PI < PI by Lm10, XXREAL_1:4;
 0 < (3 / 4) * PI by Lm10, XXREAL_1:4;
then  arccot (- 1) = (3 / 4) * PI by A1, Th18, Th36;
hence  ( arccot (- 1) = (3 / 4) * PI & arccot . (- 1) = (3 / 4) * PI ) ; ::_thesis: verum
end;

theorem Th39: :: SIN_COS9:39
( arctan 1 = PI / 4 & arctan . 1 = PI / 4 )
proof
A1: arctan 1 = arctan (tan . (PI / 4)) by SIN_COS:def_28;
 PI / 4 < PI / 2 by Lm8, XXREAL_1:4;
hence  ( arctan 1 = PI / 4 & arctan . 1 = PI / 4 ) by A1, Th35; ::_thesis: verum
end;

theorem Th40: :: SIN_COS9:40
( arccot 1 = PI / 4 & arccot . 1 = PI / 4 )
proof
A1: PI / 4 < PI by Lm9, XXREAL_1:4;
 0 < PI / 4 by Lm9, XXREAL_1:4;
then  arccot 1 = PI / 4 by A1, Th18, Th36;
hence  ( arccot 1 = PI / 4 & arccot . 1 = PI / 4 ) ; ::_thesis: verum
end;

theorem Th41: :: SIN_COS9:41
( tan . 0 = 0 & tan 0 = 0 )
proof
 PI in ].0,4.[ by SIN_COS:def_28;
then  PI > 0 by XXREAL_1:4;
then A1: PI / 2 > 0 / 2 by XREAL_1:74;
 0 - (PI / 2) < 0 by XREAL_1:49;
then A2: 0 in ].(- (PI / 2)),(PI / 2).[ by A1, XXREAL_1:4;
then tan . 0 = 0 / (cos . 0) by Th1, RFUNCT_1:def_1, SIN_COS:30
.= 0 ;
hence  ( tan . 0 = 0 & tan 0 = 0 ) by A2, Th13; ::_thesis: verum
end;

theorem Th42: :: SIN_COS9:42
( cot . (PI / 2) = 0 & cot (PI / 2) = 0 )
proof
 PI in ].0,4.[ by SIN_COS:def_28;
then A1: PI > 0 by XXREAL_1:4;
then A2: PI / 2 < PI / 1 by XREAL_1:76;
 PI / 2 > 0 / 2 by A1, XREAL_1:74;
then A3: PI / 2 in ].0,PI.[ by A2, XXREAL_1:4;
then cot . (PI / 2) = 0 / (sin . (PI / 2)) by Th2, RFUNCT_1:def_1, SIN_COS:76
.= 0 ;
hence  ( cot . (PI / 2) = 0 & cot (PI / 2) = 0 ) by A3, Th14; ::_thesis: verum
end;

theorem :: SIN_COS9:43
( arctan 0 = 0 & arctan . 0 = 0 )
proof
 PI in ].0,4.[ by SIN_COS:def_28;
then  PI > 0 by XXREAL_1:4;
then A1: PI / 2 > 0 / 2 by XREAL_1:74;
 0 - (PI / 2) < 0 by XREAL_1:49;
then  arctan 0 = 0 by A1, Th35, Th41;
hence  ( arctan 0 = 0 & arctan . 0 = 0 ) ; ::_thesis: verum
end;

theorem :: SIN_COS9:44
( arccot 0 = PI / 2 & arccot . 0 = PI / 2 )
proof
 PI in ].0,4.[ by SIN_COS:def_28;
then A1: PI > 0 by XXREAL_1:4;
then A2: PI / 2 < PI / 1 by XREAL_1:76;
 PI / 2 > 0 / 2 by A1, XREAL_1:74;
then  arccot 0 = PI / 2 by A2, Th36, Th42;
hence  ( arccot 0 = PI / 2 & arccot . 0 = PI / 2 ) ; ::_thesis: verum
end;

theorem Th45: :: SIN_COS9:45
arctan | (tan .: ].(- (PI / 2)),(PI / 2).[) is increasing
proof
set f = tan | ].(- (PI / 2)),(PI / 2).[;
A1: (tan | ].(- (PI / 2)),(PI / 2).[) .: ].(- (PI / 2)),(PI / 2).[ = rng ((tan | ].(- (PI / 2)),(PI / 2).[) | ].(- (PI / 2)),(PI / 2).[) by RELAT_1:115
.= rng (tan | ].(- (PI / 2)),(PI / 2).[) by RELAT_1:73
.= tan .: ].(- (PI / 2)),(PI / 2).[ by RELAT_1:115 ;
 (tan | ].(- (PI / 2)),(PI / 2).[) | ].(- (PI / 2)),(PI / 2).[ = tan | ].(- (PI / 2)),(PI / 2).[ by RELAT_1:73;
hence  arctan | (tan .: ].(- (PI / 2)),(PI / 2).[) is increasing by A1, Th7, FCONT_3:9; ::_thesis: verum
end;

theorem Th46: :: SIN_COS9:46
arccot | (cot .: ].0,PI.[) is decreasing
proof
set f = cot | ].0,PI.[;
A1: (cot | ].0,PI.[) .: ].0,PI.[ = rng ((cot | ].0,PI.[) | ].0,PI.[) by RELAT_1:115
.= rng (cot | ].0,PI.[) by RELAT_1:73
.= cot .: ].0,PI.[ by RELAT_1:115 ;
 (cot | ].0,PI.[) | ].0,PI.[ = cot | ].0,PI.[ by RELAT_1:73;
hence  arccot | (cot .: ].0,PI.[) is decreasing by A1, Th8, FCONT_3:10; ::_thesis: verum
end;

theorem Th47: :: SIN_COS9:47
arctan | [.(- 1),1.] is increasing
proof
A1: [.(- 1),1.] = tan .: [.(- (PI / 4)),(PI / 4).] by Th21, RELAT_1:115;
 [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12;
hence  arctan | [.(- 1),1.] is increasing by A1, Th45, RELAT_1:123, RFUNCT_2:28; ::_thesis: verum
end;

theorem Th48: :: SIN_COS9:48
arccot | [.(- 1),1.] is decreasing
proof
A1: [.(- 1),1.] = cot .: [.(PI / 4),((3 / 4) * PI).] by Th22, RELAT_1:115;
 [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12;
hence  arccot | [.(- 1),1.] is decreasing by A1, Th46, RELAT_1:123, RFUNCT_2:29; ::_thesis: verum
end;

theorem Th49: :: SIN_COS9:49
for x being set st x in [.(- 1),1.] holds
arctan . x in [.(- (PI / 4)),(PI / 4).]
proof
let x be set ; ::_thesis: ( x in [.(- 1),1.] implies arctan . x in [.(- (PI / 4)),(PI / 4).] )
assume  x in [.(- 1),1.] ; ::_thesis: arctan . x in [.(- (PI / 4)),(PI / 4).]
then  x in ].(- 1),1.[ \/ {(- 1),1} by XXREAL_1:128;
then A1: ( x in ].(- 1),1.[ or x in {(- 1),1} ) by XBOOLE_0:def_3;
percases ( x in ].(- 1),1.[ or x = - 1 or x = 1 ) by A1, TARSKI:def_2;
supposeA2: x in ].(- 1),1.[ ; ::_thesis: arctan . x in [.(- (PI / 4)),(PI / 4).]
then  x in  {  s where s is Real : ( - 1 < s & s < 1 )  }  by RCOMP_1:def_2;
then A3: ex s being Real st
( s = x & - 1 < s & s < 1 ) ;
A4: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
A5: [.(- 1),1.] /\ (dom arctan) = [.(- 1),1.] by Th23, XBOOLE_1:28;
then  1 in [.(- 1),1.] /\ (dom arctan) by XXREAL_1:1;
then A6: arctan . x < PI / 4 by A2, A5, A4, A3, Th39, Th47, RFUNCT_2:20;
 - 1 in [.(- 1),1.] by XXREAL_1:1;
then  - (PI / 4) < arctan . x by A2, A5, A4, A3, Th37, Th47, RFUNCT_2:20;
hence  arctan . x in [.(- (PI / 4)),(PI / 4).] by A6, XXREAL_1:1; ::_thesis: verum
end;
suppose x = - 1 ; ::_thesis: arctan . x in [.(- (PI / 4)),(PI / 4).]
hence  arctan . x in [.(- (PI / 4)),(PI / 4).] by Th37, XXREAL_1:1; ::_thesis: verum
end;
suppose x = 1 ; ::_thesis: arctan . x in [.(- (PI / 4)),(PI / 4).]
hence  arctan . x in [.(- (PI / 4)),(PI / 4).] by Th39, XXREAL_1:1; ::_thesis: verum
end;
end;
end;

theorem Th50: :: SIN_COS9:50
for x being set st x in [.(- 1),1.] holds
arccot . x in [.(PI / 4),((3 / 4) * PI).]
proof
let x be set ; ::_thesis: ( x in [.(- 1),1.] implies arccot . x in [.(PI / 4),((3 / 4) * PI).] )
assume  x in [.(- 1),1.] ; ::_thesis: arccot . x in [.(PI / 4),((3 / 4) * PI).]
then  x in ].(- 1),1.[ \/ {(- 1),1} by XXREAL_1:128;
then A1: ( x in ].(- 1),1.[ or x in {(- 1),1} ) by XBOOLE_0:def_3;
percases ( x in ].(- 1),1.[ or x = - 1 or x = 1 ) by A1, TARSKI:def_2;
supposeA2: x in ].(- 1),1.[ ; ::_thesis: arccot . x in [.(PI / 4),((3 / 4) * PI).]
then  x in  {  s where s is Real : ( - 1 < s & s < 1 )  }  by RCOMP_1:def_2;
then A3: ex s being Real st
( s = x & - 1 < s & s < 1 ) ;
A4: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
A5: [.(- 1),1.] /\ (dom arccot) = [.(- 1),1.] by Th24, XBOOLE_1:28;
then  1 in [.(- 1),1.] /\ (dom arccot) by XXREAL_1:1;
then A6: arccot . x > PI / 4 by A2, A5, A4, A3, Th40, Th48, RFUNCT_2:21;
 - 1 in [.(- 1),1.] by XXREAL_1:1;
then  (3 / 4) * PI > arccot . x by A2, A5, A4, A3, Th38, Th48, RFUNCT_2:21;
hence  arccot . x in [.(PI / 4),((3 / 4) * PI).] by A6, XXREAL_1:1; ::_thesis: verum
end;
supposeA7: x = - 1 ; ::_thesis: arccot . x in [.(PI / 4),((3 / 4) * PI).]
 PI in ].0,4.[ by SIN_COS:def_28;
then  PI > 0 by XXREAL_1:4;
then  PI / 4 > 0 / 4 by XREAL_1:74;
then  PI / 4 < (PI / 4) * 3 by XREAL_1:155;
hence  arccot . x in [.(PI / 4),((3 / 4) * PI).] by A7, Th38, XXREAL_1:1; ::_thesis: verum
end;
supposeA8: x = 1 ; ::_thesis: arccot . x in [.(PI / 4),((3 / 4) * PI).]
 PI in ].0,4.[ by SIN_COS:def_28;
then  PI > 0 by XXREAL_1:4;
then  PI / 4 > 0 / 4 by XREAL_1:74;
then  PI / 4 < (PI / 4) * 3 by XREAL_1:155;
hence  arccot . x in [.(PI / 4),((3 / 4) * PI).] by A8, Th40, XXREAL_1:1; ::_thesis: verum
end;
end;
end;

theorem Th51: :: SIN_COS9:51
for r being Real st - 1 <= r & r <= 1 holds
tan (arctan r) = r
proof
let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies tan (arctan r) = r )
A1: [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12;
assume that
A2: - 1 <= r and
A3: r <= 1 ; ::_thesis: tan (arctan r) = r
A4: r in [.(- 1),1.] by A2, A3, XXREAL_1:1;
then A5: r in dom (arctan | [.(- 1),1.]) by Th23, RELAT_1:62;
 arctan . r in [.(- (PI / 4)),(PI / 4).] by A4, Th49;
hence  tan (arctan r) = tan . (arctan . r) by A1, Th13
.= (tan | [.(- (PI / 4)),(PI / 4).]) . (arctan . r) by A4, Th49, FUNCT_1:49
.= (tan | [.(- (PI / 4)),(PI / 4).]) . ((arctan | [.(- 1),1.]) . r) by A4, FUNCT_1:49
.= ((tan | [.(- (PI / 4)),(PI / 4).]) * (arctan | [.(- 1),1.])) . r by A5, FUNCT_1:13
.= (id [.(- 1),1.]) . r by Th21, Th25, FUNCT_1:39
.= r by A4, FUNCT_1:18 ;
::_thesis: verum
end;

theorem Th52: :: SIN_COS9:52
for r being Real st - 1 <= r & r <= 1 holds
cot (arccot r) = r
proof
let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies cot (arccot r) = r )
A1: [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12;
assume that
A2: - 1 <= r and
A3: r <= 1 ; ::_thesis: cot (arccot r) = r
A4: r in [.(- 1),1.] by A2, A3, XXREAL_1:1;
then A5: r in dom (arccot | [.(- 1),1.]) by Th24, RELAT_1:62;
 arccot . r in [.(PI / 4),((3 / 4) * PI).] by A4, Th50;
hence  cot (arccot r) = cot . (arccot . r) by A1, Th14
.= (cot | [.(PI / 4),((3 / 4) * PI).]) . (arccot . r) by A4, Th50, FUNCT_1:49
.= (cot | [.(PI / 4),((3 / 4) * PI).]) . ((arccot | [.(- 1),1.]) . r) by A4, FUNCT_1:49
.= ((cot | [.(PI / 4),((3 / 4) * PI).]) * (arccot | [.(- 1),1.])) . r by A5, FUNCT_1:13
.= (id [.(- 1),1.]) . r by Th22, Th26, FUNCT_1:39
.= r by A4, FUNCT_1:18 ;
::_thesis: verum
end;

theorem Th53: :: SIN_COS9:53
arctan | [.(- 1),1.] is continuous
proof
set f = tan | [.(- (PI / 4)),(PI / 4).];
A1: (tan | [.(- (PI / 4)),(PI / 4).]) | [.(- (PI / 4)),(PI / 4).] = tan | [.(- (PI / 4)),(PI / 4).] by RELAT_1:72;
 (((tan | [.(- (PI / 4)),(PI / 4).]) | [.(- (PI / 4)),(PI / 4).]) ") | ((tan | [.(- (PI / 4)),(PI / 4).]) .: [.(- (PI / 4)),(PI / 4).]) is continuous by Lm11, Lm13, FCONT_1:47;
then  (arctan | [.(- 1),1.]) | [.(- 1),1.] is continuous by A1, Th21, Th25, RELAT_1:115;
hence  arctan | [.(- 1),1.] is continuous by FCONT_1:15; ::_thesis: verum
end;

theorem Th54: :: SIN_COS9:54
arccot | [.(- 1),1.] is continuous
proof
set f = cot | [.(PI / 4),((3 / 4) * PI).];
 PI in ].0,4.[ by SIN_COS:def_28;
then  PI > 0 by XXREAL_1:4;
then  PI / 4 > 0 / 4 by XREAL_1:74;
then  PI / 4 < (PI / 4) * 3 by XREAL_1:155;
then A1: (((cot | [.(PI / 4),((3 / 4) * PI).]) | [.(PI / 4),((3 / 4) * PI).]) ") | ((cot | [.(PI / 4),((3 / 4) * PI).]) .: [.(PI / 4),((3 / 4) * PI).]) is continuous by Lm12, Lm14, FCONT_1:47;
 (cot | [.(PI / 4),((3 / 4) * PI).]) | [.(PI / 4),((3 / 4) * PI).] = cot | [.(PI / 4),((3 / 4) * PI).] by RELAT_1:72;
then  (arccot | [.(- 1),1.]) | [.(- 1),1.] is continuous by A1, Th22, Th26, RELAT_1:115;
hence  arccot | [.(- 1),1.] is continuous by FCONT_1:15; ::_thesis: verum
end;

theorem Th55: :: SIN_COS9:55
rng (arctan | [.(- 1),1.]) = [.(- (PI / 4)),(PI / 4).]
proof
now__::_thesis:_for_y_being_set_holds_
(_(_y_in_[.(-_(PI_/_4)),(PI_/_4).]_implies_ex_x_being_set_st_
(_x_in_dom_(arctan_|_[.(-_1),1.])_&_y_=_(arctan_|_[.(-_1),1.])_._x_)_)_&_(_ex_x_being_set_st_
(_x_in_dom_(arctan_|_[.(-_1),1.])_&_y_=_(arctan_|_[.(-_1),1.])_._x_)_implies_y_in_[.(-_(PI_/_4)),(PI_/_4).]_)_)
let y be set ; ::_thesis: ( ( y in [.(- (PI / 4)),(PI / 4).] implies ex x being set st
( x in dom (arctan | [.(- 1),1.]) & y = (arctan | [.(- 1),1.]) . x ) ) & ( ex x being set st
( x in dom (arctan | [.(- 1),1.]) & y = (arctan | [.(- 1),1.]) . x ) implies y in [.(- (PI / 4)),(PI / 4).] ) )
thus  ( y in [.(- (PI / 4)),(PI / 4).] implies ex x being set st
( x in dom (arctan | [.(- 1),1.]) & y = (arctan | [.(- 1),1.]) . x ) ) ::_thesis: ( ex x being set st
( x in dom (arctan | [.(- 1),1.]) & y = (arctan | [.(- 1),1.]) . x ) implies y in [.(- (PI / 4)),(PI / 4).] )
proof
assume A1: y in [.(- (PI / 4)),(PI / 4).] ; ::_thesis: ex x being set st
( x in dom (arctan | [.(- 1),1.]) & y = (arctan | [.(- 1),1.]) . x )
then reconsider y1 = y as Real ;
 y1 in [.(arctan . (- 1)),(arctan . 1).] \/ [.(arctan . 1),(arctan . (- 1)).] by A1, Th37, Th39, XBOOLE_0:def_3;
then consider x being Real such that
A2: x in [.(- 1),1.] and
A3: y1 = arctan . x by Th23, Th53, FCONT_2:15;
take  x ; ::_thesis: ( x in dom (arctan | [.(- 1),1.]) & y = (arctan | [.(- 1),1.]) . x )
thus  ( x in dom (arctan | [.(- 1),1.]) & y = (arctan | [.(- 1),1.]) . x ) by A2, A3, Th23, FUNCT_1:49, RELAT_1:62; ::_thesis: verum
end;
thus  ( ex x being set st
( x in dom (arctan | [.(- 1),1.]) & y = (arctan | [.(- 1),1.]) . x ) implies y in [.(- (PI / 4)),(PI / 4).] ) ::_thesis: verum
proof
given x being set  such that A4: x in dom (arctan | [.(- 1),1.]) and
A5: y = (arctan | [.(- 1),1.]) . x ; ::_thesis: y in [.(- (PI / 4)),(PI / 4).]
A6: dom (arctan | [.(- 1),1.]) = [.(- 1),1.] by Th23, RELAT_1:62;
then  y = arctan . x by A4, A5, FUNCT_1:49;
hence  y in [.(- (PI / 4)),(PI / 4).] by A4, A6, Th49; ::_thesis: verum
end;
end;
hence  rng (arctan | [.(- 1),1.]) = [.(- (PI / 4)),(PI / 4).] by FUNCT_1:def_3; ::_thesis: verum
end;

theorem Th56: :: SIN_COS9:56
rng (arccot | [.(- 1),1.]) = [.(PI / 4),((3 / 4) * PI).]
proof
now__::_thesis:_for_y_being_set_holds_
(_(_y_in_[.(PI_/_4),((3_/_4)_*_PI).]_implies_ex_x_being_set_st_
(_x_in_dom_(arccot_|_[.(-_1),1.])_&_y_=_(arccot_|_[.(-_1),1.])_._x_)_)_&_(_ex_x_being_set_st_
(_x_in_dom_(arccot_|_[.(-_1),1.])_&_y_=_(arccot_|_[.(-_1),1.])_._x_)_implies_y_in_[.(PI_/_4),((3_/_4)_*_PI).]_)_)
let y be set ; ::_thesis: ( ( y in [.(PI / 4),((3 / 4) * PI).] implies ex x being set st
( x in dom (arccot | [.(- 1),1.]) & y = (arccot | [.(- 1),1.]) . x ) ) & ( ex x being set st
( x in dom (arccot | [.(- 1),1.]) & y = (arccot | [.(- 1),1.]) . x ) implies y in [.(PI / 4),((3 / 4) * PI).] ) )
thus  ( y in [.(PI / 4),((3 / 4) * PI).] implies ex x being set st
( x in dom (arccot | [.(- 1),1.]) & y = (arccot | [.(- 1),1.]) . x ) ) ::_thesis: ( ex x being set st
( x in dom (arccot | [.(- 1),1.]) & y = (arccot | [.(- 1),1.]) . x ) implies y in [.(PI / 4),((3 / 4) * PI).] )
proof
assume A1: y in [.(PI / 4),((3 / 4) * PI).] ; ::_thesis: ex x being set st
( x in dom (arccot | [.(- 1),1.]) & y = (arccot | [.(- 1),1.]) . x )
then reconsider y1 = y as Real ;
 y1 in [.(arccot . 1),(arccot . (- 1)).] \/ [.(arccot . (- 1)),(arccot . 1).] by A1, Th38, Th40, XBOOLE_0:def_3;
then consider x being Real such that
A2: x in [.(- 1),1.] and
A3: y1 = arccot . x by Th24, Th54, FCONT_2:15;
take  x ; ::_thesis: ( x in dom (arccot | [.(- 1),1.]) & y = (arccot | [.(- 1),1.]) . x )
thus  ( x in dom (arccot | [.(- 1),1.]) & y = (arccot | [.(- 1),1.]) . x ) by A2, A3, Th24, FUNCT_1:49, RELAT_1:62; ::_thesis: verum
end;
thus  ( ex x being set st
( x in dom (arccot | [.(- 1),1.]) & y = (arccot | [.(- 1),1.]) . x ) implies y in [.(PI / 4),((3 / 4) * PI).] ) ::_thesis: verum
proof
given x being set  such that A4: x in dom (arccot | [.(- 1),1.]) and
A5: y = (arccot | [.(- 1),1.]) . x ; ::_thesis: y in [.(PI / 4),((3 / 4) * PI).]
A6: dom (arccot | [.(- 1),1.]) = [.(- 1),1.] by Th24, RELAT_1:62;
then  y = arccot . x by A4, A5, FUNCT_1:49;
hence  y in [.(PI / 4),((3 / 4) * PI).] by A4, A6, Th50; ::_thesis: verum
end;
end;
hence  rng (arccot | [.(- 1),1.]) = [.(PI / 4),((3 / 4) * PI).] by FUNCT_1:def_3; ::_thesis: verum
end;

theorem :: SIN_COS9:57
for r being Real st - 1 <= r & r <= 1 & arctan r = - (PI / 4) holds
r = - 1 by Th17, Th51;

theorem :: SIN_COS9:58
for r being Real st - 1 <= r & r <= 1 & arccot r = (3 / 4) * PI holds
r = - 1 by Th18, Th52;

theorem :: SIN_COS9:59
for r being Real st - 1 <= r & r <= 1 & arctan r = 0 holds
r = 0 by Th41, Th51;

theorem :: SIN_COS9:60
for r being Real st - 1 <= r & r <= 1 & arccot r = PI / 2 holds
r = 0 by Th42, Th52;

theorem :: SIN_COS9:61
for r being Real st - 1 <= r & r <= 1 & arctan r = PI / 4 holds
r = 1
proof
let r be Real; ::_thesis: ( - 1 <= r & r <= 1 & arctan r = PI / 4 implies r = 1 )
assume that
A1: - 1 <= r and
A2: r <= 1 and
A3: arctan r = PI / 4 ; ::_thesis: r = 1
thus r = tan (PI / 4) by A1, A2, A3, Th51
.= tan . (PI / 4) by Lm8, Th13
.= 1 by SIN_COS:def_28 ; ::_thesis: verum
end;

theorem :: SIN_COS9:62
for r being Real st - 1 <= r & r <= 1 & arccot r = PI / 4 holds
r = 1 by Th18, Th52;

theorem Th63: :: SIN_COS9:63
for r being Real st - 1 <= r & r <= 1 holds
( - (PI / 4) <= arctan r & arctan r <= PI / 4 )
proof
let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies ( - (PI / 4) <= arctan r & arctan r <= PI / 4 ) )
assume that
A1: - 1 <= r and
A2: r <= 1 ; ::_thesis: ( - (PI / 4) <= arctan r & arctan r <= PI / 4 )
A3: r in [.(- 1),1.] by A1, A2, XXREAL_1:1;
then  r in dom (arctan | [.(- 1),1.]) by Th23, RELAT_1:62;
then  (arctan | [.(- 1),1.]) . r in rng (arctan | [.(- 1),1.]) by FUNCT_1:def_3;
then  arctan r in rng (arctan | [.(- 1),1.]) by A3, FUNCT_1:49;
hence  ( - (PI / 4) <= arctan r & arctan r <= PI / 4 ) by Th55, XXREAL_1:1; ::_thesis: verum
end;

theorem Th64: :: SIN_COS9:64
for r being Real st - 1 <= r & r <= 1 holds
( PI / 4 <= arccot r & arccot r <= (3 / 4) * PI )
proof
let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies ( PI / 4 <= arccot r & arccot r <= (3 / 4) * PI ) )
assume that
A1: - 1 <= r and
A2: r <= 1 ; ::_thesis: ( PI / 4 <= arccot r & arccot r <= (3 / 4) * PI )
A3: r in [.(- 1),1.] by A1, A2, XXREAL_1:1;
then  r in dom (arccot | [.(- 1),1.]) by Th24, RELAT_1:62;
then  (arccot | [.(- 1),1.]) . r in rng (arccot | [.(- 1),1.]) by FUNCT_1:def_3;
then  arccot r in rng (arccot | [.(- 1),1.]) by A3, FUNCT_1:49;
hence  ( PI / 4 <= arccot r & arccot r <= (3 / 4) * PI ) by Th56, XXREAL_1:1; ::_thesis: verum
end;

theorem :: SIN_COS9:65
for r being Real st - 1 < r & r < 1 holds
( - (PI / 4) < arctan r & arctan r < PI / 4 )
proof
let r be Real; ::_thesis: ( - 1 < r & r < 1 implies ( - (PI / 4) < arctan r & arctan r < PI / 4 ) )
A1: tan (PI / 4) = tan . (PI / 4) by Lm8, Th13
.= 1 by SIN_COS:def_28 ;
assume that
A2: - 1 < r and
A3: r < 1 ; ::_thesis: ( - (PI / 4) < arctan r & arctan r < PI / 4 )
A4: arctan r <= PI / 4 by A2, A3, Th63;
 - (PI / 4) <= arctan r by A2, A3, Th63;
then  ( ( - (PI / 4) < arctan r & arctan r < PI / 4 ) or - (PI / 4) = arctan r or arctan r = PI / 4 ) by A4, XXREAL_0:1;
hence  ( - (PI / 4) < arctan r & arctan r < PI / 4 ) by A2, A3, A1, Th17, Th51; ::_thesis: verum
end;

theorem :: SIN_COS9:66
for r being Real st - 1 < r & r < 1 holds
( PI / 4 < arccot r & arccot r < (3 / 4) * PI )
proof
let r be Real; ::_thesis: ( - 1 < r & r < 1 implies ( PI / 4 < arccot r & arccot r < (3 / 4) * PI ) )
assume that
A1: - 1 < r and
A2: r < 1 ; ::_thesis: ( PI / 4 < arccot r & arccot r < (3 / 4) * PI )
A3: arccot r <= (3 / 4) * PI by A1, A2, Th64;
 PI / 4 <= arccot r by A1, A2, Th64;
then  ( ( PI / 4 < arccot r & arccot r < (3 / 4) * PI ) or PI / 4 = arccot r or arccot r = (3 / 4) * PI ) by A3, XXREAL_0:1;
hence  ( PI / 4 < arccot r & arccot r < (3 / 4) * PI ) by A1, A2, Th18, Th52; ::_thesis: verum
end;

theorem :: SIN_COS9:67
for r being Real st - 1 <= r & r <= 1 holds
arctan r = - (arctan (- r))
proof
let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies arctan r = - (arctan (- r)) )
A1: [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12;
assume that
A2: - 1 <= r and
A3: r <= 1 ; ::_thesis: arctan r = - (arctan (- r))
A4: - r >= - 1 by A3, XREAL_1:24;
A5: - (- 1) >= - r by A2, XREAL_1:24;
then A6: arctan (- r) <= PI / 4 by A4, Th63;
 - (PI / 4) <= arctan (- r) by A5, A4, Th63;
then A7: - (arctan (- r)) <= - (- (PI / 4)) by XREAL_1:24;
 arctan (- r) <= PI / 4 by A5, A4, Th63;
then  - (PI / 4) <= - (arctan (- r)) by XREAL_1:24;
then A8: - (arctan (- r)) in [.(- (PI / 4)),(PI / 4).] by A7, XXREAL_1:1;
 - (PI / 4) <= arctan (- r) by A5, A4, Th63;
then  arctan (- r) in [.(- (PI / 4)),(PI / 4).] by A6, XXREAL_1:1;
then A9: cos (arctan (- r)) <> 0 by A1, COMPTRIG:11;
 tan (arctan (- r)) = - r by A5, A4, Th51;
then A10: r = ((tan 0) - (tan (arctan (- r)))) / (1 + ((tan 0) * (tan (arctan (- r))))) by Th41
.= tan (0 - (arctan (- r))) by A9, SIN_COS:31, SIN_COS4:8 ;
A11: [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12;
then A12: - (arctan (- r)) < PI / 2 by A8, XXREAL_1:4;
 - (PI / 2) < - (arctan (- r)) by A8, A11, XXREAL_1:4;
hence  arctan r = - (arctan (- r)) by A10, A12, Th35; ::_thesis: verum
end;

theorem :: SIN_COS9:68
for r being Real st - 1 <= r & r <= 1 holds
arccot r = PI - (arccot (- r))
proof
let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies arccot r = PI - (arccot (- r)) )
set x = arccot (- r);
assume that
A1: - 1 <= r and
A2: r <= 1 ; ::_thesis: arccot r = PI - (arccot (- r))
A3: - r >= - 1 by A2, XREAL_1:24;
A4: - (- 1) >= - r by A1, XREAL_1:24;
then  - r = cot (arccot (- r)) by A3, Th52;
then A5: r = - (cot (arccot (- r)))
.= - ((cos (arccot (- r))) / (sin (arccot (- r)))) by SIN_COS4:def_2
.= (cos (arccot (- r))) / (- (sin (arccot (- r)))) by XCMPLX_1:188
.= (cos (arccot (- r))) / (sin (- (arccot (- r)))) by SIN_COS:31
.= (cos (- (arccot (- r)))) / (sin (- (arccot (- r)))) by SIN_COS:31
.= cot (- (arccot (- r))) by SIN_COS4:def_2 ;
 - r in [.(- 1),1.] by A4, A3, XXREAL_1:1;
then A6: arccot (- r) in [.(PI / 4),((3 / 4) * PI).] by Th50;
then  arccot (- r) <= (3 / 4) * PI by XXREAL_1:1;
then  - (arccot (- r)) >= - ((3 / 4) * PI) by XREAL_1:24;
then A7: PI + (- (arccot (- r))) >= PI + (- ((3 / 4) * PI)) by XREAL_1:6;
 PI / 4 <= arccot (- r) by A6, XXREAL_1:1;
then  - (PI / 4) >= - (arccot (- r)) by XREAL_1:24;
then  PI + (- (PI / 4)) >= PI + (- (arccot (- r))) by XREAL_1:6;
then A8: PI + (- (arccot (- r))) in [.(PI / 4),((3 / 4) * PI).] by A7, XXREAL_1:1;
A9: [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12;
then A10: PI + (- (arccot (- r))) < PI by A8, XXREAL_1:4;
A11: cot (PI + (- (arccot (- r)))) = (cos (PI + (- (arccot (- r))))) / (sin (PI + (- (arccot (- r))))) by SIN_COS4:def_2
.= (- (cos (- (arccot (- r))))) / (sin (PI + (- (arccot (- r))))) by SIN_COS:79
.= (- (cos (- (arccot (- r))))) / (- (sin (- (arccot (- r))))) by SIN_COS:79
.= (cos (- (arccot (- r)))) / (sin (- (arccot (- r)))) by XCMPLX_1:191
.= cot (- (arccot (- r))) by SIN_COS4:def_2 ;
 0 < PI + (- (arccot (- r))) by A8, A9, XXREAL_1:4;
hence  arccot r = PI - (arccot (- r)) by A5, A10, A11, Th36; ::_thesis: verum
end;

theorem :: SIN_COS9:69
for r being Real st - 1 <= r & r <= 1 holds
cot (arctan r) = 1 / r
proof
let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies cot (arctan r) = 1 / r )
set x = arctan r;
assume that
A1: - 1 <= r and
A2: r <= 1 ; ::_thesis: cot (arctan r) = 1 / r
A3: (sin (arctan r)) / (cos (arctan r)) = tan (arctan r) by SIN_COS4:def_1
.= r by A1, A2, Th51 ;
cot (arctan r) = (cos (arctan r)) / (sin (arctan r)) by SIN_COS4:def_2
.= 1 / r by A3, XCMPLX_1:57 ;
hence  cot (arctan r) = 1 / r ; ::_thesis: verum
end;

theorem :: SIN_COS9:70
for r being Real st - 1 <= r & r <= 1 holds
tan (arccot r) = 1 / r
proof
let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies tan (arccot r) = 1 / r )
set x = arccot r;
assume that
A1: - 1 <= r and
A2: r <= 1 ; ::_thesis: tan (arccot r) = 1 / r
A3: (cos (arccot r)) / (sin (arccot r)) = cot (arccot r) by SIN_COS4:def_2
.= r by A1, A2, Th52 ;
tan (arccot r) = (sin (arccot r)) / (cos (arccot r)) by SIN_COS4:def_1
.= 1 / r by A3, XCMPLX_1:57 ;
hence  tan (arccot r) = 1 / r ; ::_thesis: verum
end;

theorem Th71: :: SIN_COS9:71
arctan is_differentiable_on tan .: ].(- (PI / 2)),(PI / 2).[
proof
set f = tan | ].(- (PI / 2)),(PI / 2).[;
A1: dom ((tan | ].(- (PI / 2)),(PI / 2).[) ") = rng (tan | ].(- (PI / 2)),(PI / 2).[) by FUNCT_1:33
.= tan .: ].(- (PI / 2)),(PI / 2).[ by RELAT_1:115 ;
 dom (tan | ].(- (PI / 2)),(PI / 2).[) = (dom tan) /\ ].(- (PI / 2)),(PI / 2).[ by RELAT_1:61;
then A2: ].(- (PI / 2)),(PI / 2).[ c= dom (tan | ].(- (PI / 2)),(PI / 2).[) by Th1, XBOOLE_1:19;
A3: tan | ].(- (PI / 2)),(PI / 2).[ is_differentiable_on ].(- (PI / 2)),(PI / 2).[ by Lm1, FDIFF_2:16;
A4: now__::_thesis:_for_x0_being_Real_st_x0_in_].(-_(PI_/_2)),(PI_/_2).[_holds_
0_<_diff_((tan_|_].(-_(PI_/_2)),(PI_/_2).[),x0)
A5: for x0 being Real st x0 in ].(- (PI / 2)),(PI / 2).[ holds
1 / ((cos . x0) ^2) > 0
proof
let x0 be Real; ::_thesis: ( x0 in ].(- (PI / 2)),(PI / 2).[ implies 1 / ((cos . x0) ^2) > 0 )
assume  x0 in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: 1 / ((cos . x0) ^2) > 0
then  0 < cos . x0 by COMPTRIG:11;
then  (cos . x0) ^2 > 0 by SQUARE_1:12;
then  1 / ((cos . x0) ^2) > 0 / ((cos . x0) ^2) by XREAL_1:74;
hence  1 / ((cos . x0) ^2) > 0 ; ::_thesis: verum
end;
let x0 be Real; ::_thesis: ( x0 in ].(- (PI / 2)),(PI / 2).[ implies 0 < diff ((tan | ].(- (PI / 2)),(PI / 2).[),x0) )
assume A6: x0 in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: 0 < diff ((tan | ].(- (PI / 2)),(PI / 2).[),x0)
diff ((tan | ].(- (PI / 2)),(PI / 2).[),x0) = ((tan | ].(- (PI / 2)),(PI / 2).[) `| ].(- (PI / 2)),(PI / 2).[) . x0 by A3, A6, FDIFF_1:def_7
.= (tan `| ].(- (PI / 2)),(PI / 2).[) . x0 by Lm1, FDIFF_2:16
.= diff (tan,x0) by A6, Lm1, FDIFF_1:def_7
.= 1 / ((cos . x0) ^2) by A6, Lm3 ;
hence  0 < diff ((tan | ].(- (PI / 2)),(PI / 2).[),x0) by A6, A5; ::_thesis: verum
end;
 (tan | ].(- (PI / 2)),(PI / 2).[) | ].(- (PI / 2)),(PI / 2).[ = tan | ].(- (PI / 2)),(PI / 2).[ by RELAT_1:72;
hence  arctan is_differentiable_on tan .: ].(- (PI / 2)),(PI / 2).[ by A1, A3, A2, A4, FDIFF_2:48; ::_thesis: verum
end;

theorem Th72: :: SIN_COS9:72
arccot is_differentiable_on cot .: ].0,PI.[
proof
set f = cot | ].0,PI.[;
A1: dom ((cot | ].0,PI.[) ") = rng (cot | ].0,PI.[) by FUNCT_1:33
.= cot .: ].0,PI.[ by RELAT_1:115 ;
 dom (cot | ].0,PI.[) = (dom cot) /\ ].0,PI.[ by RELAT_1:61;
then A2: ].0,PI.[ c= dom (cot | ].0,PI.[) by Th2, XBOOLE_1:19;
A3: cot | ].0,PI.[ is_differentiable_on ].0,PI.[ by Lm2, FDIFF_2:16;
A4: now__::_thesis:_for_x0_being_Real_st_x0_in_].0,PI.[_holds_
diff_((cot_|_].0,PI.[),x0)_<_0
A5: for x0 being Real st x0 in ].0,PI.[ holds
- (1 / ((sin . x0) ^2)) < 0
proof
let x0 be Real; ::_thesis: ( x0 in ].0,PI.[ implies - (1 / ((sin . x0) ^2)) < 0 )
assume  x0 in ].0,PI.[ ; ::_thesis: - (1 / ((sin . x0) ^2)) < 0
then  0 < sin . x0 by COMPTRIG:7;
then  (sin . x0) ^2 > 0 by SQUARE_1:12;
then  1 / ((sin . x0) ^2) > 0 / ((sin . x0) ^2) by XREAL_1:74;
then  - (1 / ((sin . x0) ^2)) < - 0 by XREAL_1:24;
hence  - (1 / ((sin . x0) ^2)) < 0 ; ::_thesis: verum
end;
let x0 be Real; ::_thesis: ( x0 in ].0,PI.[ implies diff ((cot | ].0,PI.[),x0) < 0 )
assume A6: x0 in ].0,PI.[ ; ::_thesis: diff ((cot | ].0,PI.[),x0) < 0
diff ((cot | ].0,PI.[),x0) = ((cot | ].0,PI.[) `| ].0,PI.[) . x0 by A3, A6, FDIFF_1:def_7
.= (cot `| ].0,PI.[) . x0 by Lm2, FDIFF_2:16
.= diff (cot,x0) by A6, Lm2, FDIFF_1:def_7
.= - (1 / ((sin . x0) ^2)) by A6, Lm4 ;
hence  diff ((cot | ].0,PI.[),x0) < 0 by A6, A5; ::_thesis: verum
end;
 (cot | ].0,PI.[) | ].0,PI.[ = cot | ].0,PI.[ by RELAT_1:72;
hence  arccot is_differentiable_on cot .: ].0,PI.[ by A1, A2, A3, A4, FDIFF_2:48; ::_thesis: verum
end;

theorem Th73: :: SIN_COS9:73
arctan is_differentiable_on ].(- 1),1.[
proof
 ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then A1: ].(- 1),1.[ c= dom arctan by Th23, XBOOLE_1:1;
 for x being Real st x in ].(- 1),1.[ holds
arctan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in ].(- 1),1.[ implies arctan is_differentiable_in x )
A2: dom arctan = rng (tan | ].(- (PI / 2)),(PI / 2).[) by FUNCT_1:33
.= tan .: ].(- (PI / 2)),(PI / 2).[ by RELAT_1:115 ;
assume  x in ].(- 1),1.[ ; ::_thesis: arctan is_differentiable_in x
then  arctan | (dom arctan) is_differentiable_in x by A1, A2, Th71, FDIFF_1:def_6;
hence  arctan is_differentiable_in x by RELAT_1:69; ::_thesis: verum
end;
hence  arctan is_differentiable_on ].(- 1),1.[ by A1, FDIFF_1:9; ::_thesis: verum
end;

theorem Th74: :: SIN_COS9:74
arccot is_differentiable_on ].(- 1),1.[
proof
 ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then A1: ].(- 1),1.[ c= dom arccot by Th24, XBOOLE_1:1;
 for x being Real st x in ].(- 1),1.[ holds
arccot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in ].(- 1),1.[ implies arccot is_differentiable_in x )
A2: dom arccot = rng (cot | ].0,PI.[) by FUNCT_1:33
.= cot .: ].0,PI.[ by RELAT_1:115 ;
assume  x in ].(- 1),1.[ ; ::_thesis: arccot is_differentiable_in x
then  arccot | (dom arccot) is_differentiable_in x by A1, A2, Th72, FDIFF_1:def_6;
hence  arccot is_differentiable_in x by RELAT_1:69; ::_thesis: verum
end;
hence  arccot is_differentiable_on ].(- 1),1.[ by A1, FDIFF_1:9; ::_thesis: verum
end;

theorem Th75: :: SIN_COS9:75
for r being Real st - 1 <= r & r <= 1 holds
diff (arctan,r) = 1 / (1 + (r ^2))
proof
let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies diff (arctan,r) = 1 / (1 + (r ^2)) )
set g = arctan ;
set f = tan | ].(- (PI / 2)),(PI / 2).[;
set x = arctan . r;
assume that
A1: - 1 <= r and
A2: r <= 1 ; ::_thesis: diff (arctan,r) = 1 / (1 + (r ^2))
A3: ((sin . (arctan . r)) ^2) + ((cos . (arctan . r)) ^2) = 1 by SIN_COS:28;
A4: tan | ].(- (PI / 2)),(PI / 2).[ is_differentiable_on ].(- (PI / 2)),(PI / 2).[ by Lm1, FDIFF_2:16;
A5: now__::_thesis:_for_x0_being_Real_st_x0_in_].(-_(PI_/_2)),(PI_/_2).[_holds_
0_<_diff_((tan_|_].(-_(PI_/_2)),(PI_/_2).[),x0)
A6: for x0 being Real st x0 in ].(- (PI / 2)),(PI / 2).[ holds
1 / ((cos . x0) ^2) > 0
proof
let x0 be Real; ::_thesis: ( x0 in ].(- (PI / 2)),(PI / 2).[ implies 1 / ((cos . x0) ^2) > 0 )
assume  x0 in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: 1 / ((cos . x0) ^2) > 0
then  0 < cos . x0 by COMPTRIG:11;
then  (cos . x0) ^2 > 0 by SQUARE_1:12;
then  1 / ((cos . x0) ^2) > 0 / ((cos . x0) ^2) by XREAL_1:74;
hence  1 / ((cos . x0) ^2) > 0 ; ::_thesis: verum
end;
let x0 be Real; ::_thesis: ( x0 in ].(- (PI / 2)),(PI / 2).[ implies 0 < diff ((tan | ].(- (PI / 2)),(PI / 2).[),x0) )
assume A7: x0 in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: 0 < diff ((tan | ].(- (PI / 2)),(PI / 2).[),x0)
diff ((tan | ].(- (PI / 2)),(PI / 2).[),x0) = ((tan | ].(- (PI / 2)),(PI / 2).[) `| ].(- (PI / 2)),(PI / 2).[) . x0 by A4, A7, FDIFF_1:def_7
.= (tan `| ].(- (PI / 2)),(PI / 2).[) . x0 by Lm1, FDIFF_2:16
.= diff (tan,x0) by A7, Lm1, FDIFF_1:def_7
.= 1 / ((cos . x0) ^2) by A7, Lm3 ;
hence  0 < diff ((tan | ].(- (PI / 2)),(PI / 2).[),x0) by A7, A6; ::_thesis: verum
end;
A8: r in [.(- 1),1.] by A1, A2, XXREAL_1:1;
then A9: arctan . r in [.(- (PI / 4)),(PI / 4).] by Th49;
 arctan . r = arctan r ;
then A10: r = tan (arctan . r) by A1, A2, Th51
.= (sin (arctan . r)) / (cos (arctan . r)) by SIN_COS4:def_1 ;
 dom (tan | ].(- (PI / 2)),(PI / 2).[) = (dom tan) /\ ].(- (PI / 2)),(PI / 2).[ by RELAT_1:61;
then A11: ].(- (PI / 2)),(PI / 2).[ c= dom (tan | ].(- (PI / 2)),(PI / 2).[) by Th1, XBOOLE_1:19;
A12: (tan | ].(- (PI / 2)),(PI / 2).[) | ].(- (PI / 2)),(PI / 2).[ = tan | ].(- (PI / 2)),(PI / 2).[ by RELAT_1:72;
A13: [.(- (PI / 4)),(PI / 4).] c= ].(- (PI / 2)),(PI / 2).[ by Lm7, Lm8, XXREAL_2:def_12;
then  cos (arctan . r) <> 0 by A9, COMPTRIG:11;
then  r * (cos (arctan . r)) = sin (arctan . r) by A10, XCMPLX_1:87;
then A14: 1 = ((cos (arctan . r)) ^2) * ((r ^2) + 1) by A3;
 tan | ].(- (PI / 2)),(PI / 2).[ is_differentiable_on ].(- (PI / 2)),(PI / 2).[ by Lm1, FDIFF_2:16;
then  diff ((tan | ].(- (PI / 2)),(PI / 2).[),(arctan . r)) = ((tan | ].(- (PI / 2)),(PI / 2).[) `| ].(- (PI / 2)),(PI / 2).[) . (arctan . r) by A9, A13, FDIFF_1:def_7
.= (tan `| ].(- (PI / 2)),(PI / 2).[) . (arctan . r) by Lm1, FDIFF_2:16
.= diff (tan,(arctan . r)) by A9, A13, Lm1, FDIFF_1:def_7
.= 1 / ((cos (arctan . r)) ^2) by A9, A13, Lm3 ;
then  diff (arctan,r) = 1 / (1 / ((cos (arctan . r)) ^2)) by A8, A4, A5, A12, A11, Th23, FDIFF_2:48
.= 1 / ((r ^2) + 1) by A14, XCMPLX_1:73 ;
hence  diff (arctan,r) = 1 / (1 + (r ^2)) ; ::_thesis: verum
end;

theorem Th76: :: SIN_COS9:76
for r being Real st - 1 <= r & r <= 1 holds
diff (arccot,r) = - (1 / (1 + (r ^2)))
proof
let r be Real; ::_thesis: ( - 1 <= r & r <= 1 implies diff (arccot,r) = - (1 / (1 + (r ^2))) )
set g = arccot ;
set f = cot | ].0,PI.[;
set x = arccot . r;
assume that
A1: - 1 <= r and
A2: r <= 1 ; ::_thesis: diff (arccot,r) = - (1 / (1 + (r ^2)))
A3: ((sin . (arccot . r)) ^2) + ((cos . (arccot . r)) ^2) = 1 by SIN_COS:28;
A4: cot | ].0,PI.[ is_differentiable_on ].0,PI.[ by Lm2, FDIFF_2:16;
A5: now__::_thesis:_for_x0_being_Real_st_x0_in_].0,PI.[_holds_
diff_((cot_|_].0,PI.[),x0)_<_0
A6: for x0 being Real st x0 in ].0,PI.[ holds
- (1 / ((sin . x0) ^2)) < 0
proof
let x0 be Real; ::_thesis: ( x0 in ].0,PI.[ implies - (1 / ((sin . x0) ^2)) < 0 )
assume  x0 in ].0,PI.[ ; ::_thesis: - (1 / ((sin . x0) ^2)) < 0
then  0 < sin . x0 by COMPTRIG:7;
then  (sin . x0) ^2 > 0 by SQUARE_1:12;
then  1 / ((sin . x0) ^2) > 0 / ((sin . x0) ^2) by XREAL_1:74;
then  - (1 / ((sin . x0) ^2)) < - 0 by XREAL_1:24;
hence  - (1 / ((sin . x0) ^2)) < 0 ; ::_thesis: verum
end;
let x0 be Real; ::_thesis: ( x0 in ].0,PI.[ implies diff ((cot | ].0,PI.[),x0) < 0 )
assume A7: x0 in ].0,PI.[ ; ::_thesis: diff ((cot | ].0,PI.[),x0) < 0
diff ((cot | ].0,PI.[),x0) = ((cot | ].0,PI.[) `| ].0,PI.[) . x0 by A4, A7, FDIFF_1:def_7
.= (cot `| ].0,PI.[) . x0 by Lm2, FDIFF_2:16
.= diff (cot,x0) by A7, Lm2, FDIFF_1:def_7
.= - (1 / ((sin . x0) ^2)) by A7, Lm4 ;
hence  diff ((cot | ].0,PI.[),x0) < 0 by A7, A6; ::_thesis: verum
end;
A8: r in [.(- 1),1.] by A1, A2, XXREAL_1:1;
then A9: arccot . r in [.(PI / 4),((3 / 4) * PI).] by Th50;
 arccot . r = arccot r ;
then A10: r = cot (arccot . r) by A1, A2, Th52
.= (cos (arccot . r)) / (sin (arccot . r)) by SIN_COS4:def_2 ;
 dom (cot | ].0,PI.[) = (dom cot) /\ ].0,PI.[ by RELAT_1:61;
then A11: ].0,PI.[ c= dom (cot | ].0,PI.[) by Th2, XBOOLE_1:19;
A12: (cot | ].0,PI.[) | ].0,PI.[ = cot | ].0,PI.[ by RELAT_1:72;
A13: [.(PI / 4),((3 / 4) * PI).] c= ].0,PI.[ by Lm9, Lm10, XXREAL_2:def_12;
then  sin (arccot . r) <> 0 by A9, COMPTRIG:7;
then  r * (sin (arccot . r)) = cos (arccot . r) by A10, XCMPLX_1:87;
then A14: 1 = ((sin (arccot . r)) ^2) * ((r ^2) + 1) by A3;
 cot | ].0,PI.[ is_differentiable_on ].0,PI.[ by Lm2, FDIFF_2:16;
then  diff ((cot | ].0,PI.[),(arccot . r)) = ((cot | ].0,PI.[) `| ].0,PI.[) . (arccot . r) by A9, A13, FDIFF_1:def_7
.= (cot `| ].0,PI.[) . (arccot . r) by Lm2, FDIFF_2:16
.= diff (cot,(arccot . r)) by A9, A13, Lm2, FDIFF_1:def_7
.= - (1 / ((sin (arccot . r)) ^2)) by A9, A13, Lm4 ;
then  diff (arccot,r) = 1 / (- (1 / ((sin (arccot . r)) ^2))) by A8, A4, A5, A12, A11, Th24, FDIFF_2:48
.= - (1 / (1 / ((sin (arccot . r)) ^2))) by XCMPLX_1:188
.= - (1 / ((r ^2) + 1)) by A14, XCMPLX_1:73 ;
hence  diff (arccot,r) = - (1 / (1 + (r ^2))) ; ::_thesis: verum
end;

theorem :: SIN_COS9:77
arctan | (tan .: ].(- (PI / 2)),(PI / 2).[) is continuous by Th71, FDIFF_1:25;

theorem :: SIN_COS9:78
arccot | (cot .: ].0,PI.[) is continuous by Th72, FDIFF_1:25;

theorem :: SIN_COS9:79
dom arctan is open
proof
 for x0 being Real st x0 in ].(- (PI / 2)),(PI / 2).[ holds
0 < diff (tan,x0)
proof
let x0 be Real; ::_thesis: ( x0 in ].(- (PI / 2)),(PI / 2).[ implies 0 < diff (tan,x0) )
assume A1: x0 in ].(- (PI / 2)),(PI / 2).[ ; ::_thesis: 0 < diff (tan,x0)
then  0 < cos . x0 by COMPTRIG:11;
then  (cos . x0) ^2 > 0 by SQUARE_1:12;
then  1 / ((cos . x0) ^2) > 0 / ((cos . x0) ^2) by XREAL_1:74;
hence  0 < diff (tan,x0) by A1, Lm3; ::_thesis: verum
end;
then  rng (tan | ].(- (PI / 2)),(PI / 2).[) is open by Lm1, Th1, FDIFF_2:41;
hence  dom arctan is open by FUNCT_1:33; ::_thesis: verum
end;

theorem :: SIN_COS9:80
dom arccot is open
proof
 for x0 being Real st x0 in ].0,PI.[ holds
diff (cot,x0) < 0
proof
let x0 be Real; ::_thesis: ( x0 in ].0,PI.[ implies diff (cot,x0) < 0 )
assume A1: x0 in ].0,PI.[ ; ::_thesis: diff (cot,x0) < 0
then  0 < sin . x0 by COMPTRIG:7;
then  (sin . x0) ^2 > 0 by SQUARE_1:12;
then  1 / ((sin . x0) ^2) > 0 / ((sin . x0) ^2) by XREAL_1:74;
then  - (1 / ((sin . x0) ^2)) < - 0 by XREAL_1:24;
hence  diff (cot,x0) < 0 by A1, Lm4; ::_thesis: verum
end;
then  rng (cot | ].0,PI.[) is open by Lm2, Th2, FDIFF_2:41;
hence  dom arccot is open by FUNCT_1:33; ::_thesis: verum
end;

begin

theorem Th81: :: SIN_COS9:81
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(arctan `| Z) . x = 1 / (1 + (x ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(arctan `| Z) . x = 1 / (1 + (x ^2)) ) ) )
assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(arctan `| Z) . x = 1 / (1 + (x ^2)) ) )
then A2: arctan is_differentiable_on Z by Th73, FDIFF_1:26;
 for x being Real st x in Z holds
(arctan `| Z) . x = 1 / (1 + (x ^2))
proof
let x be Real; ::_thesis: ( x in Z implies (arctan `| Z) . x = 1 / (1 + (x ^2)) )
assume A3: x in Z ; ::_thesis: (arctan `| Z) . x = 1 / (1 + (x ^2))
then A4: - 1 <= x by A1, XXREAL_1:4;
A5: x <= 1 by A1, A3, XXREAL_1:4;
thus (arctan `| Z) . x = diff (arctan,x) by A2, A3, FDIFF_1:def_7
.= 1 / (1 + (x ^2)) by A4, A5, Th75 ; ::_thesis: verum
end;
hence  ( arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(arctan `| Z) . x = 1 / (1 + (x ^2)) ) ) by A1, Th73, FDIFF_1:26; ::_thesis: verum
end;

theorem Th82: :: SIN_COS9:82
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(arccot `| Z) . x = - (1 / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(arccot `| Z) . x = - (1 / (1 + (x ^2))) ) ) )
assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(arccot `| Z) . x = - (1 / (1 + (x ^2))) ) )
then A2: arccot is_differentiable_on Z by Th74, FDIFF_1:26;
 for x being Real st x in Z holds
(arccot `| Z) . x = - (1 / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies (arccot `| Z) . x = - (1 / (1 + (x ^2))) )
assume A3: x in Z ; ::_thesis: (arccot `| Z) . x = - (1 / (1 + (x ^2)))
then A4: - 1 <= x by A1, XXREAL_1:4;
A5: x <= 1 by A1, A3, XXREAL_1:4;
thus (arccot `| Z) . x = diff (arccot,x) by A2, A3, FDIFF_1:def_7
.= - (1 / (1 + (x ^2))) by A4, A5, Th76 ; ::_thesis: verum
end;
hence  ( arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(arccot `| Z) . x = - (1 / (1 + (x ^2))) ) ) by A1, Th74, FDIFF_1:26; ::_thesis: verum
end;

theorem :: SIN_COS9:83
for r being Real
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( r (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ) )
proof
let r be Real; ::_thesis: for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( r (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ) )
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( r (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ) ) )
assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( r (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ) )
 ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arctan by Th23, XBOOLE_1:1;
then  Z c= dom arctan by A1, XBOOLE_1:1;
then A2: Z c= dom (r (#) arctan) by VALUED_1:def_5;
A3: arctan is_differentiable_on Z by A1, Th81;
 for x being Real st x in Z holds
((r (#) arctan) `| Z) . x = r / (1 + (x ^2))
proof
let x be Real; ::_thesis: ( x in Z implies ((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) )
assume A4: x in Z ; ::_thesis: ((r (#) arctan) `| Z) . x = r / (1 + (x ^2))
then A5: - 1 < x by A1, XXREAL_1:4;
A6: x < 1 by A1, A4, XXREAL_1:4;
((r (#) arctan) `| Z) . x = r * (diff (arctan,x)) by A2, A3, A4, FDIFF_1:20
.= r * (1 / (1 + (x ^2))) by A5, A6, Th75
.= r / (1 + (x ^2)) ;
hence  ((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ; ::_thesis: verum
end;
hence  ( r (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ) ) by A2, A3, FDIFF_1:20; ::_thesis: verum
end;

theorem :: SIN_COS9:84
for r being Real
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( r (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2))) ) )
proof
let r be Real; ::_thesis: for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( r (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2))) ) )
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( r (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2))) ) ) )
assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( r (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2))) ) )
 ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by Th24, XBOOLE_1:1;
then  Z c= dom arccot by A1, XBOOLE_1:1;
then A2: Z c= dom (r (#) arccot) by VALUED_1:def_5;
A3: arccot is_differentiable_on Z by A1, Th82;
 for x being Real st x in Z holds
((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2))) )
assume A4: x in Z ; ::_thesis: ((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2)))
then A5: - 1 < x by A1, XXREAL_1:4;
A6: x < 1 by A1, A4, XXREAL_1:4;
((r (#) arccot) `| Z) . x = r * (diff (arccot,x)) by A2, A3, A4, FDIFF_1:20
.= r * (- (1 / (1 + (x ^2)))) by A5, A6, Th76
.= - (r / (1 + (x ^2))) ;
hence  ((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( r (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arccot) `| Z) . x = - (r / (1 + (x ^2))) ) ) by A2, A3, FDIFF_1:20; ::_thesis: verum
end;

theorem Th85: :: SIN_COS9:85
for x being Real
for f being PartFunc of REAL,REAL st f is_differentiable_in x & f . x > - 1 & f . x < 1 holds
( arctan * f is_differentiable_in x & diff ((arctan * f),x) = (diff (f,x)) / (1 + ((f . x) ^2)) )
proof
let x be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_differentiable_in x & f . x > - 1 & f . x < 1 holds
( arctan * f is_differentiable_in x & diff ((arctan * f),x) = (diff (f,x)) / (1 + ((f . x) ^2)) )
let f be PartFunc of REAL,REAL; ::_thesis: ( f is_differentiable_in x & f . x > - 1 & f . x < 1 implies ( arctan * f is_differentiable_in x & diff ((arctan * f),x) = (diff (f,x)) / (1 + ((f . x) ^2)) ) )
assume that
A1: f is_differentiable_in x and
A2: f . x > - 1 and
A3: f . x < 1 ; ::_thesis: ( arctan * f is_differentiable_in x & diff ((arctan * f),x) = (diff (f,x)) / (1 + ((f . x) ^2)) )
 f . x in ].(- 1),1.[ by A2, A3, XXREAL_1:4;
then A4: arctan is_differentiable_in f . x by Th73, FDIFF_1:9;
then  diff ((arctan * f),x) = (diff (arctan,(f . x))) * (diff (f,x)) by A1, FDIFF_2:13
.= (diff (f,x)) * (1 / (1 + ((f . x) ^2))) by A2, A3, Th75
.= (diff (f,x)) / (1 + ((f . x) ^2)) ;
hence  ( arctan * f is_differentiable_in x & diff ((arctan * f),x) = (diff (f,x)) / (1 + ((f . x) ^2)) ) by A1, A4, FDIFF_2:13; ::_thesis: verum
end;

theorem Th86: :: SIN_COS9:86
for x being Real
for f being PartFunc of REAL,REAL st f is_differentiable_in x & f . x > - 1 & f . x < 1 holds
( arccot * f is_differentiable_in x & diff ((arccot * f),x) = - ((diff (f,x)) / (1 + ((f . x) ^2))) )
proof
let x be Real; ::_thesis: for f being PartFunc of REAL,REAL st f is_differentiable_in x & f . x > - 1 & f . x < 1 holds
( arccot * f is_differentiable_in x & diff ((arccot * f),x) = - ((diff (f,x)) / (1 + ((f . x) ^2))) )
let f be PartFunc of REAL,REAL; ::_thesis: ( f is_differentiable_in x & f . x > - 1 & f . x < 1 implies ( arccot * f is_differentiable_in x & diff ((arccot * f),x) = - ((diff (f,x)) / (1 + ((f . x) ^2))) ) )
assume that
A1: f is_differentiable_in x and
A2: f . x > - 1 and
A3: f . x < 1 ; ::_thesis: ( arccot * f is_differentiable_in x & diff ((arccot * f),x) = - ((diff (f,x)) / (1 + ((f . x) ^2))) )
 f . x in ].(- 1),1.[ by A2, A3, XXREAL_1:4;
then A4: arccot is_differentiable_in f . x by Th74, FDIFF_1:9;
then  diff ((arccot * f),x) = (diff (arccot,(f . x))) * (diff (f,x)) by A1, FDIFF_2:13
.= (diff (f,x)) * (- (1 / (1 + ((f . x) ^2)))) by A2, A3, Th76
.= - ((diff (f,x)) / (1 + ((f . x) ^2))) ;
hence  ( arccot * f is_differentiable_in x & diff ((arccot * f),x) = - ((diff (f,x)) / (1 + ((f . x) ^2))) ) by A1, A4, FDIFF_2:13; ::_thesis: verum
end;

theorem Th87: :: SIN_COS9:87
for r, s being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (arctan * f) & ( for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) holds
( arctan * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) ) )
proof
let r, s be Real; ::_thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (arctan * f) & ( for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) holds
( arctan * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) ) )
let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (arctan * f) & ( for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) holds
( arctan * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) ) )
let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (arctan * f) & ( for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) implies ( arctan * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) ) ) )
assume that
A1: Z c= dom (arctan * f) and
A2: for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ; ::_thesis: ( arctan * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) ) )
 for y being set st y in Z holds
y in dom f by A1, FUNCT_1:11;
then A3: Z c= dom f by TARSKI:def_3;
A4: for x being Real st x in Z holds
f . x = (r * x) + s by A2;
then A5: f is_differentiable_on Z by A3, FDIFF_1:23;
A6: for x being Real st x in Z holds
arctan * f is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arctan * f is_differentiable_in x )
assume A7: x in Z ; ::_thesis: arctan * f is_differentiable_in x
then A8: f . x > - 1 by A2;
A9: f . x < 1 by A2, A7;
 f is_differentiable_in x by A5, A7, FDIFF_1:9;
hence  arctan * f is_differentiable_in x by A8, A9, Th85; ::_thesis: verum
end;
then A10: arctan * f is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2))
proof
let x be Real; ::_thesis: ( x in Z implies ((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) )
assume A11: x in Z ; ::_thesis: ((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2))
then A12: f . x > - 1 by A2;
A13: f . x < 1 by A2, A11;
 f is_differentiable_in x by A5, A11, FDIFF_1:9;
then  diff ((arctan * f),x) = (diff (f,x)) / (1 + ((f . x) ^2)) by A12, A13, Th85
.= ((f `| Z) . x) / (1 + ((f . x) ^2)) by A5, A11, FDIFF_1:def_7
.= r / (1 + ((f . x) ^2)) by A4, A3, A11, FDIFF_1:23
.= r / (1 + (((r * x) + s) ^2)) by A2, A11 ;
hence  ((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) by A10, A11, FDIFF_1:def_7; ::_thesis: verum
end;
hence  ( arctan * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * f) `| Z) . x = r / (1 + (((r * x) + s) ^2)) ) ) by A1, A6, FDIFF_1:9; ::_thesis: verum
end;

theorem Th88: :: SIN_COS9:88
for r, s being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (arccot * f) & ( for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) holds
( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) )
proof
let r, s be Real; ::_thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (arccot * f) & ( for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) holds
( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) )
let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (arccot * f) & ( for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) holds
( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) )
let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (arccot * f) & ( for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) implies ( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) ) )
assume that
A1: Z c= dom (arccot * f) and
A2: for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ; ::_thesis: ( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) )
 for y being set st y in Z holds
y in dom f by A1, FUNCT_1:11;
then A3: Z c= dom f by TARSKI:def_3;
A4: for x being Real st x in Z holds
f . x = (r * x) + s by A2;
then A5: f is_differentiable_on Z by A3, FDIFF_1:23;
A6: for x being Real st x in Z holds
arccot * f is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arccot * f is_differentiable_in x )
assume A7: x in Z ; ::_thesis: arccot * f is_differentiable_in x
then A8: f . x > - 1 by A2;
A9: f . x < 1 by A2, A7;
 f is_differentiable_in x by A5, A7, FDIFF_1:9;
hence  arccot * f is_differentiable_in x by A8, A9, Th86; ::_thesis: verum
end;
then A10: arccot * f is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) )
assume A11: x in Z ; ::_thesis: ((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2)))
then A12: f . x > - 1 by A2;
A13: f . x < 1 by A2, A11;
 f is_differentiable_in x by A5, A11, FDIFF_1:9;
then  diff ((arccot * f),x) = - ((diff (f,x)) / (1 + ((f . x) ^2))) by A12, A13, Th86
.= - (((f `| Z) . x) / (1 + ((f . x) ^2))) by A5, A11, FDIFF_1:def_7
.= - (r / (1 + ((f . x) ^2))) by A4, A3, A11, FDIFF_1:23
.= - (r / (1 + (((r * x) + s) ^2))) by A2, A11 ;
hence  ((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) by A10, A11, FDIFF_1:def_7; ::_thesis: verum
end;
hence  ( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) ) by A1, A6, FDIFF_1:9; ::_thesis: verum
end;

theorem :: SIN_COS9:89
for Z being open Subset of REAL st Z c= dom (ln * arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
arctan . x > 0 ) holds
( ln * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (ln * arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
arctan . x > 0 ) implies ( ln * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) ) ) )
assume that
A1: Z c= dom (ln * arctan) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
arctan . x > 0 ; ::_thesis: ( ln * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) ) )
A4: for x being Real st x in Z holds
ln * arctan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies ln * arctan is_differentiable_in x )
assume A5: x in Z ; ::_thesis: ln * arctan is_differentiable_in x
 arctan is_differentiable_on Z by A2, Th81;
then A6: arctan is_differentiable_in x by A5, FDIFF_1:9;
 arctan . x > 0 by A3, A5;
hence  ln * arctan is_differentiable_in x by A6, TAYLOR_1:20; ::_thesis: verum
end;
then A7: ln * arctan is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x))
proof
let x be Real; ::_thesis: ( x in Z implies ((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) )
assume A8: x in Z ; ::_thesis: ((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x))
then A9: - 1 < x by A2, XXREAL_1:4;
 arctan is_differentiable_on Z by A2, Th81;
then A10: arctan is_differentiable_in x by A8, FDIFF_1:9;
A11: x < 1 by A2, A8, XXREAL_1:4;
 arctan . x > 0 by A3, A8;
then  diff ((ln * arctan),x) = (diff (arctan,x)) / (arctan . x) by A10, TAYLOR_1:20
.= (1 / (1 + (x ^2))) / (arctan . x) by A9, A11, Th75
.= 1 / ((1 + (x ^2)) * (arctan . x)) by XCMPLX_1:78 ;
hence  ((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) by A7, A8, FDIFF_1:def_7; ::_thesis: verum
end;
hence  ( ln * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum
end;

theorem :: SIN_COS9:90
for Z being open Subset of REAL st Z c= dom (ln * arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
arccot . x > 0 ) holds
( ln * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (ln * arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
arccot . x > 0 ) implies ( ln * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x))) ) ) )
assume that
A1: Z c= dom (ln * arccot) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
arccot . x > 0 ; ::_thesis: ( ln * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x))) ) )
A4: for x being Real st x in Z holds
ln * arccot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies ln * arccot is_differentiable_in x )
assume A5: x in Z ; ::_thesis: ln * arccot is_differentiable_in x
 arccot is_differentiable_on Z by A2, Th82;
then A6: arccot is_differentiable_in x by A5, FDIFF_1:9;
 arccot . x > 0 by A3, A5;
hence  ln * arccot is_differentiable_in x by A6, TAYLOR_1:20; ::_thesis: verum
end;
then A7: ln * arccot is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x)))
proof
let x be Real; ::_thesis: ( x in Z implies ((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x))) )
assume A8: x in Z ; ::_thesis: ((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x)))
then A9: - 1 < x by A2, XXREAL_1:4;
 arccot is_differentiable_on Z by A2, Th82;
then A10: arccot is_differentiable_in x by A8, FDIFF_1:9;
A11: x < 1 by A2, A8, XXREAL_1:4;
 arccot . x > 0 by A3, A8;
then  diff ((ln * arccot),x) = (diff (arccot,x)) / (arccot . x) by A10, TAYLOR_1:20
.= (- (1 / (1 + (x ^2)))) / (arccot . x) by A9, A11, Th76
.= - ((1 / (1 + (x ^2))) / (arccot . x))
.= - (1 / ((1 + (x ^2)) * (arccot . x))) by XCMPLX_1:78 ;
hence  ((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x))) by A7, A8, FDIFF_1:def_7; ::_thesis: verum
end;
hence  ( ln * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (arccot . x))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum
end;

theorem Th91: :: SIN_COS9:91
for n being Element of NAT
for Z being open Subset of REAL st Z c= dom ((#Z n) * arctan) & Z c= ].(- 1),1.[ holds
( (#Z n) * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) ) )
proof
let n be Element of NAT ; ::_thesis: for Z being open Subset of REAL st Z c= dom ((#Z n) * arctan) & Z c= ].(- 1),1.[ holds
( (#Z n) * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) ) )
let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((#Z n) * arctan) & Z c= ].(- 1),1.[ implies ( (#Z n) * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) ) ) )
assume that
A1: Z c= dom ((#Z n) * arctan) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( (#Z n) * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) ) )
A3: for x being Real st x in Z holds
(#Z n) * arctan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies (#Z n) * arctan is_differentiable_in x )
assume A4: x in Z ; ::_thesis: (#Z n) * arctan is_differentiable_in x
 arctan is_differentiable_on Z by A2, Th81;
then  arctan is_differentiable_in x by A4, FDIFF_1:9;
hence  (#Z n) * arctan is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum
end;
then A5: (#Z n) * arctan is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
(((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2))
proof
let x be Real; ::_thesis: ( x in Z implies (((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) )
assume A6: x in Z ; ::_thesis: (((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2))
then A7: - 1 < x by A2, XXREAL_1:4;
 arctan is_differentiable_on Z by A2, Th81;
then A8: arctan is_differentiable_in x by A6, FDIFF_1:9;
A9: x < 1 by A2, A6, XXREAL_1:4;
(((#Z n) * arctan) `| Z) . x = diff (((#Z n) * arctan),x) by A5, A6, FDIFF_1:def_7
.= (n * ((arctan . x) #Z (n - 1))) * (diff (arctan,x)) by A8, TAYLOR_1:3
.= (n * ((arctan . x) #Z (n - 1))) * (1 / (1 + (x ^2))) by A7, A9, Th75
.= (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) ;
hence  (((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) ; ::_thesis: verum
end;
hence  ( (#Z n) * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z n) * arctan) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2)) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem Th92: :: SIN_COS9:92
for n being Element of NAT
for Z being open Subset of REAL st Z c= dom ((#Z n) * arccot) & Z c= ].(- 1),1.[ holds
( (#Z n) * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) ) )
proof
let n be Element of NAT ; ::_thesis: for Z being open Subset of REAL st Z c= dom ((#Z n) * arccot) & Z c= ].(- 1),1.[ holds
( (#Z n) * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) ) )
let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((#Z n) * arccot) & Z c= ].(- 1),1.[ implies ( (#Z n) * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom ((#Z n) * arccot) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( (#Z n) * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) ) )
A3: for x being Real st x in Z holds
(#Z n) * arccot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies (#Z n) * arccot is_differentiable_in x )
assume A4: x in Z ; ::_thesis: (#Z n) * arccot is_differentiable_in x
 arccot is_differentiable_on Z by A2, Th82;
then  arccot is_differentiable_in x by A4, FDIFF_1:9;
hence  (#Z n) * arccot is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum
end;
then A5: (#Z n) * arccot is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
(((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies (((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) )
assume A6: x in Z ; ::_thesis: (((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2)))
then A7: - 1 < x by A2, XXREAL_1:4;
 arccot is_differentiable_on Z by A2, Th82;
then A8: arccot is_differentiable_in x by A6, FDIFF_1:9;
A9: x < 1 by A2, A6, XXREAL_1:4;
(((#Z n) * arccot) `| Z) . x = diff (((#Z n) * arccot),x) by A5, A6, FDIFF_1:def_7
.= (n * ((arccot . x) #Z (n - 1))) * (diff (arccot,x)) by A8, TAYLOR_1:3
.= (n * ((arccot . x) #Z (n - 1))) * (- (1 / (1 + (x ^2)))) by A7, A9, Th76
.= - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) ;
hence  (((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( (#Z n) * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z n) * arccot) `| Z) . x = - ((n * ((arccot . x) #Z (n - 1))) / (1 + (x ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem :: SIN_COS9:93
for Z being open Subset of REAL st Z c= dom ((1 / 2) (#) ((#Z 2) * arctan)) & Z c= ].(- 1),1.[ holds
( (1 / 2) (#) ((#Z 2) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (arctan . x) / (1 + (x ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((1 / 2) (#) ((#Z 2) * arctan)) & Z c= ].(- 1),1.[ implies ( (1 / 2) (#) ((#Z 2) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (arctan . x) / (1 + (x ^2)) ) ) )
assume that
A1: Z c= dom ((1 / 2) (#) ((#Z 2) * arctan)) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( (1 / 2) (#) ((#Z 2) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (arctan . x) / (1 + (x ^2)) ) )
A3: Z c= dom ((#Z 2) * arctan) by A1, VALUED_1:def_5;
then A4: (#Z 2) * arctan is_differentiable_on Z by A2, Th91;
 for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (arctan . x) / (1 + (x ^2))
proof
let x be Real; ::_thesis: ( x in Z implies (((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (arctan . x) / (1 + (x ^2)) )
assume A5: x in Z ; ::_thesis: (((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (arctan . x) / (1 + (x ^2))
then (((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (1 / 2) * (diff (((#Z 2) * arctan),x)) by A1, A4, FDIFF_1:20
.= (1 / 2) * ((((#Z 2) * arctan) `| Z) . x) by A4, A5, FDIFF_1:def_7
.= (1 / 2) * ((2 * ((arctan . x) #Z (2 - 1))) / (1 + (x ^2))) by A2, A3, A5, Th91
.= (1 / 2) * ((2 * (arctan . x)) / (1 + (x ^2))) by PREPOWER:35
.= (arctan . x) / (1 + (x ^2)) ;
hence  (((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (arctan . x) / (1 + (x ^2)) ; ::_thesis: verum
end;
hence  ( (1 / 2) (#) ((#Z 2) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x = (arctan . x) / (1 + (x ^2)) ) ) by A1, A4, FDIFF_1:20; ::_thesis: verum
end;

theorem :: SIN_COS9:94
for Z being open Subset of REAL st Z c= dom ((1 / 2) (#) ((#Z 2) * arccot)) & Z c= ].(- 1),1.[ holds
( (1 / 2) (#) ((#Z 2) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((1 / 2) (#) ((#Z 2) * arccot)) & Z c= ].(- 1),1.[ implies ( (1 / 2) (#) ((#Z 2) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom ((1 / 2) (#) ((#Z 2) * arccot)) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( (1 / 2) (#) ((#Z 2) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2))) ) )
A3: Z c= dom ((#Z 2) * arccot) by A1, VALUED_1:def_5;
then A4: (#Z 2) * arccot is_differentiable_on Z by A2, Th92;
 for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2))) )
assume A5: x in Z ; ::_thesis: (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2)))
then (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = (1 / 2) * (diff (((#Z 2) * arccot),x)) by A1, A4, FDIFF_1:20
.= (1 / 2) * ((((#Z 2) * arccot) `| Z) . x) by A4, A5, FDIFF_1:def_7
.= (1 / 2) * (- ((2 * ((arccot . x) #Z (2 - 1))) / (1 + (x ^2)))) by A2, A3, A5, Th92
.= - ((1 / 2) * ((2 * ((arccot . x) #Z 1)) / (1 + (x ^2))))
.= - ((1 / 2) * ((2 * (arccot . x)) / (1 + (x ^2)))) by PREPOWER:35
.= - ((arccot . x) / (1 + (x ^2))) ;
hence  (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( (1 / 2) (#) ((#Z 2) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x = - ((arccot . x) / (1 + (x ^2))) ) ) by A1, A4, FDIFF_1:20; ::_thesis: verum
end;

theorem Th95: :: SIN_COS9:95
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( (id Z) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) arctan) `| Z) . x = (arctan . x) + (x / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( (id Z) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) arctan) `| Z) . x = (arctan . x) + (x / (1 + (x ^2))) ) ) )
assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( (id Z) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) arctan) `| Z) . x = (arctan . x) + (x / (1 + (x ^2))) ) )
A2: Z c= dom (id Z) by FUNCT_1:17;
 ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arctan by Th23, XBOOLE_1:1;
then  Z c= dom arctan by A1, XBOOLE_1:1;
then  Z c= (dom (id Z)) /\ (dom arctan) by A2, XBOOLE_1:19;
then A3: Z c= dom ((id Z) (#) arctan) by VALUED_1:def_4;
A4: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
then A5: id Z is_differentiable_on Z by A2, FDIFF_1:23;
A6: arctan is_differentiable_on Z by A1, Th81;
 for x being Real st x in Z holds
(((id Z) (#) arctan) `| Z) . x = (arctan . x) + (x / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies (((id Z) (#) arctan) `| Z) . x = (arctan . x) + (x / (1 + (x ^2))) )
assume A7: x in Z ; ::_thesis: (((id Z) (#) arctan) `| Z) . x = (arctan . x) + (x / (1 + (x ^2)))
then A8: - 1 < x by A1, XXREAL_1:4;
A9: x < 1 by A1, A7, XXREAL_1:4;
(((id Z) (#) arctan) `| Z) . x = ((arctan . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff (arctan,x))) by A3, A5, A6, A7, FDIFF_1:21
.= ((arctan . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff (arctan,x))) by A5, A7, FDIFF_1:def_7
.= ((arctan . x) * 1) + (((id Z) . x) * (diff (arctan,x))) by A2, A4, A7, FDIFF_1:23
.= (arctan . x) + (x * (diff (arctan,x))) by A7, FUNCT_1:18
.= (arctan . x) + (x * (1 / (1 + (x ^2)))) by A8, A9, Th75
.= (arctan . x) + (x / (1 + (x ^2))) ;
hence  (((id Z) (#) arctan) `| Z) . x = (arctan . x) + (x / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( (id Z) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) arctan) `| Z) . x = (arctan . x) + (x / (1 + (x ^2))) ) ) by A3, A5, A6, FDIFF_1:21; ::_thesis: verum
end;

theorem Th96: :: SIN_COS9:96
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( (id Z) (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) arccot) `| Z) . x = (arccot . x) - (x / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( (id Z) (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) arccot) `| Z) . x = (arccot . x) - (x / (1 + (x ^2))) ) ) )
assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( (id Z) (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) arccot) `| Z) . x = (arccot . x) - (x / (1 + (x ^2))) ) )
A2: Z c= dom (id Z) by FUNCT_1:17;
 ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by Th24, XBOOLE_1:1;
then  Z c= dom arccot by A1, XBOOLE_1:1;
then  Z c= (dom (id Z)) /\ (dom arccot) by A2, XBOOLE_1:19;
then A3: Z c= dom ((id Z) (#) arccot) by VALUED_1:def_4;
A4: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
then A5: id Z is_differentiable_on Z by A2, FDIFF_1:23;
A6: arccot is_differentiable_on Z by A1, Th82;
 for x being Real st x in Z holds
(((id Z) (#) arccot) `| Z) . x = (arccot . x) - (x / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies (((id Z) (#) arccot) `| Z) . x = (arccot . x) - (x / (1 + (x ^2))) )
assume A7: x in Z ; ::_thesis: (((id Z) (#) arccot) `| Z) . x = (arccot . x) - (x / (1 + (x ^2)))
then A8: - 1 < x by A1, XXREAL_1:4;
A9: x < 1 by A1, A7, XXREAL_1:4;
(((id Z) (#) arccot) `| Z) . x = ((arccot . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff (arccot,x))) by A3, A5, A6, A7, FDIFF_1:21
.= ((arccot . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff (arccot,x))) by A5, A7, FDIFF_1:def_7
.= ((arccot . x) * 1) + (((id Z) . x) * (diff (arccot,x))) by A2, A4, A7, FDIFF_1:23
.= (arccot . x) + (x * (diff (arccot,x))) by A7, FUNCT_1:18
.= (arccot . x) + (x * (- (1 / (1 + (x ^2))))) by A8, A9, Th76
.= (arccot . x) - (x / (1 + (x ^2))) ;
hence  (((id Z) (#) arccot) `| Z) . x = (arccot . x) - (x / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( (id Z) (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) arccot) `| Z) . x = (arccot . x) - (x / (1 + (x ^2))) ) ) by A3, A5, A6, FDIFF_1:21; ::_thesis: verum
end;

theorem :: SIN_COS9:97
for r, s being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (f (#) arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (r * x) + s ) holds
( f (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) ) )
proof
let r, s be Real; ::_thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (f (#) arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (r * x) + s ) holds
( f (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) ) )
let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (f (#) arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (r * x) + s ) holds
( f (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) ) )
let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (f (#) arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (r * x) + s ) implies ( f (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (f (#) arctan) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
f . x = (r * x) + s ; ::_thesis: ( f (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) ) )
 Z c= (dom f) /\ (dom arctan) by A1, VALUED_1:def_4;
then A4: Z c= dom f by XBOOLE_1:18;
then A5: f is_differentiable_on Z by A3, FDIFF_1:23;
A6: arctan is_differentiable_on Z by A2, Th81;
 for x being Real st x in Z holds
((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) )
assume A7: x in Z ; ::_thesis: ((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2)))
then A8: - 1 < x by A2, XXREAL_1:4;
A9: x < 1 by A2, A7, XXREAL_1:4;
((f (#) arctan) `| Z) . x = ((arctan . x) * (diff (f,x))) + ((f . x) * (diff (arctan,x))) by A1, A5, A6, A7, FDIFF_1:21
.= ((arctan . x) * ((f `| Z) . x)) + ((f . x) * (diff (arctan,x))) by A5, A7, FDIFF_1:def_7
.= ((arctan . x) * r) + ((f . x) * (diff (arctan,x))) by A3, A4, A7, FDIFF_1:23
.= ((arctan . x) * r) + ((f . x) * (1 / (1 + (x ^2)))) by A8, A9, Th75
.= (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) by A3, A7 ;
hence  ((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( f (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arctan) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2))) ) ) by A1, A5, A6, FDIFF_1:21; ::_thesis: verum
end;

theorem :: SIN_COS9:98
for r, s being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (f (#) arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (r * x) + s ) holds
( f (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) ) )
proof
let r, s be Real; ::_thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (f (#) arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (r * x) + s ) holds
( f (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) ) )
let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (f (#) arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (r * x) + s ) holds
( f (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) ) )
let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (f (#) arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (r * x) + s ) implies ( f (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (f (#) arccot) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
f . x = (r * x) + s ; ::_thesis: ( f (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) ) )
 Z c= (dom f) /\ (dom arccot) by A1, VALUED_1:def_4;
then A4: Z c= dom f by XBOOLE_1:18;
then A5: f is_differentiable_on Z by A3, FDIFF_1:23;
A6: arccot is_differentiable_on Z by A2, Th82;
 for x being Real st x in Z holds
((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) )
assume A7: x in Z ; ::_thesis: ((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2)))
then A8: - 1 < x by A2, XXREAL_1:4;
A9: x < 1 by A2, A7, XXREAL_1:4;
((f (#) arccot) `| Z) . x = ((arccot . x) * (diff (f,x))) + ((f . x) * (diff (arccot,x))) by A1, A5, A6, A7, FDIFF_1:21
.= ((arccot . x) * ((f `| Z) . x)) + ((f . x) * (diff (arccot,x))) by A5, A7, FDIFF_1:def_7
.= ((arccot . x) * r) + ((f . x) * (diff (arccot,x))) by A3, A4, A7, FDIFF_1:23
.= ((arccot . x) * r) + ((f . x) * (- (1 / (1 + (x ^2))))) by A8, A9, Th76
.= ((arccot . x) * r) - ((f . x) * (1 / (1 + (x ^2))))
.= (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) by A3, A7 ;
hence  ((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( f (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arccot) `| Z) . x = (r * (arccot . x)) - (((r * x) + s) / (1 + (x ^2))) ) ) by A1, A5, A6, FDIFF_1:21; ::_thesis: verum
end;

theorem :: SIN_COS9:99
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (arctan * f)) & ( for x being Real st x in Z holds
( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ) holds
( (1 / 2) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (arctan * f)) & ( for x being Real st x in Z holds
( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ) holds
( (1 / 2) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2)) ) )
let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / 2) (#) (arctan * f)) & ( for x being Real st x in Z holds
( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ) implies ( (1 / 2) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2)) ) ) )
assume that
A1: Z c= dom ((1 / 2) (#) (arctan * f)) and
A2: for x being Real st x in Z holds
( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ; ::_thesis: ( (1 / 2) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2)) ) )
A3: for x being Real st x in Z holds
( f . x = (2 * x) + 0 & f . x > - 1 & f . x < 1 ) by A2;
A4: Z c= dom (arctan * f) by A1, VALUED_1:def_5;
then A5: arctan * f is_differentiable_on Z by A3, Th87;
 for x being Real st x in Z holds
(((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2))
proof
let x be Real; ::_thesis: ( x in Z implies (((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2)) )
assume A6: x in Z ; ::_thesis: (((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2))
then (((1 / 2) (#) (arctan * f)) `| Z) . x = (1 / 2) * (diff ((arctan * f),x)) by A1, A5, FDIFF_1:20
.= (1 / 2) * (((arctan * f) `| Z) . x) by A5, A6, FDIFF_1:def_7
.= (1 / 2) * (2 / (1 + (((2 * x) + 0) ^2))) by A4, A3, A6, Th87
.= 1 / (1 + ((2 * x) ^2)) ;
hence  (((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2)) ; ::_thesis: verum
end;
hence  ( (1 / 2) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (arctan * f)) `| Z) . x = 1 / (1 + ((2 * x) ^2)) ) ) by A1, A5, FDIFF_1:20; ::_thesis: verum
end;

theorem :: SIN_COS9:100
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (arccot * f)) & ( for x being Real st x in Z holds
( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ) holds
( (1 / 2) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (arccot * f)) & ( for x being Real st x in Z holds
( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ) holds
( (1 / 2) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2))) ) )
let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / 2) (#) (arccot * f)) & ( for x being Real st x in Z holds
( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ) implies ( (1 / 2) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2))) ) ) )
assume that
A1: Z c= dom ((1 / 2) (#) (arccot * f)) and
A2: for x being Real st x in Z holds
( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ; ::_thesis: ( (1 / 2) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2))) ) )
A3: for x being Real st x in Z holds
( f . x = (2 * x) + 0 & f . x > - 1 & f . x < 1 ) by A2;
A4: Z c= dom (arccot * f) by A1, VALUED_1:def_5;
then A5: arccot * f is_differentiable_on Z by A3, Th88;
 for x being Real st x in Z holds
(((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies (((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2))) )
assume A6: x in Z ; ::_thesis: (((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2)))
then (((1 / 2) (#) (arccot * f)) `| Z) . x = (1 / 2) * (diff ((arccot * f),x)) by A1, A5, FDIFF_1:20
.= (1 / 2) * (((arccot * f) `| Z) . x) by A5, A6, FDIFF_1:def_7
.= (1 / 2) * (- (2 / (1 + (((2 * x) + 0) ^2)))) by A4, A3, A6, Th88
.= - (1 / (1 + ((2 * x) ^2))) ;
hence  (((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2))) ; ::_thesis: verum
end;
hence  ( (1 / 2) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (arccot * f)) `| Z) . x = - (1 / (1 + ((2 * x) ^2))) ) ) by A1, A5, FDIFF_1:20; ::_thesis: verum
end;

theorem Th101: :: SIN_COS9:101
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = #Z 2 holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x ) )
proof
let Z be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = #Z 2 holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x ) )
let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = #Z 2 implies ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x ) ) )
assume that
A1: Z c= dom (f1 + f2) and
A2: for x being Real st x in Z holds
f1 . x = 1 and
A3: f2 = #Z 2 ; ::_thesis: ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x ) )
A4: for x being Real st x in Z holds
f2 is_differentiable_in x by A3, TAYLOR_1:2;
A5: Z c= (dom f1) /\ (dom f2) by A1, VALUED_1:def_1;
then A6: Z c= dom f1 by XBOOLE_1:18;
A7: for x being Real st x in Z holds
f1 . x = (0 * x) + 1 by A2;
then A8: f1 is_differentiable_on Z by A6, FDIFF_1:23;
 Z c= dom f2 by A5, XBOOLE_1:18;
then A9: f2 is_differentiable_on Z by A4, FDIFF_1:9;
A10: for x being Real st x in Z holds
(f2 `| Z) . x = 2 * x
proof
let x be Real; ::_thesis: ( x in Z implies (f2 `| Z) . x = 2 * x )
 2 * (x #Z (2 - 1)) = 2 * x by PREPOWER:35;
then A11: diff (f2,x) = 2 * x by A3, TAYLOR_1:2;
assume  x in Z ; ::_thesis: (f2 `| Z) . x = 2 * x
hence  (f2 `| Z) . x = 2 * x by A9, A11, FDIFF_1:def_7; ::_thesis: verum
end;
 for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x
proof
let x be Real; ::_thesis: ( x in Z implies ((f1 + f2) `| Z) . x = 2 * x )
assume A12: x in Z ; ::_thesis: ((f1 + f2) `| Z) . x = 2 * x
then ((f1 + f2) `| Z) . x = (diff (f1,x)) + (diff (f2,x)) by A1, A8, A9, FDIFF_1:18
.= ((f1 `| Z) . x) + (diff (f2,x)) by A8, A12, FDIFF_1:def_7
.= ((f1 `| Z) . x) + ((f2 `| Z) . x) by A9, A12, FDIFF_1:def_7
.= 0 + ((f2 `| Z) . x) by A6, A7, A12, FDIFF_1:23
.= 2 * x by A10, A12 ;
hence  ((f1 + f2) `| Z) . x = 2 * x ; ::_thesis: verum
end;
hence  ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x ) ) by A1, A8, A9, FDIFF_1:18; ::_thesis: verum
end;

theorem Th102: :: SIN_COS9:102
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (ln * (f1 + f2))) & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (ln * (f1 + f2))) & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) ) )
let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((1 / 2) (#) (ln * (f1 + f2))) & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) implies ( (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) ) ) )
assume that
A1: Z c= dom ((1 / 2) (#) (ln * (f1 + f2))) and
A2: f2 = #Z 2 and
A3: for x being Real st x in Z holds
f1 . x = 1 ; ::_thesis: ( (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) ) )
A4: Z c= dom (ln * (f1 + f2)) by A1, VALUED_1:def_5;
then  for y being set st y in Z holds
y in dom (f1 + f2) by FUNCT_1:11;
then A5: Z c= dom (f1 + f2) by TARSKI:def_3;
then A6: f1 + f2 is_differentiable_on Z by A2, A3, Th101;
 for x being Real st x in Z holds
ln * (f1 + f2) is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies ln * (f1 + f2) is_differentiable_in x )
assume A7: x in Z ; ::_thesis: ln * (f1 + f2) is_differentiable_in x
then (f1 + f2) . x = (f1 . x) + (f2 . x) by A5, VALUED_1:def_1
.= 1 + (f2 . x) by A3, A7
.= 1 + (x #Z (1 + 1)) by A2, TAYLOR_1:def_1
.= 1 + ((x #Z 1) * (x #Z 1)) by TAYLOR_1:1
.= 1 + (x * (x #Z 1)) by PREPOWER:35
.= 1 + (x * x) by PREPOWER:35 ;
then A8: (f1 + f2) . x > 0 by XREAL_1:34, XREAL_1:63;
 f1 + f2 is_differentiable_in x by A6, A7, FDIFF_1:9;
hence  ln * (f1 + f2) is_differentiable_in x by A8, TAYLOR_1:20; ::_thesis: verum
end;
then A9: ln * (f1 + f2) is_differentiable_on Z by A4, FDIFF_1:9;
 for x being Real st x in Z holds
(((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2))
proof
let x be Real; ::_thesis: ( x in Z implies (((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) )
assume A10: x in Z ; ::_thesis: (((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2))
then A11: f1 + f2 is_differentiable_in x by A6, FDIFF_1:9;
A12: (f1 + f2) . x = (f1 . x) + (f2 . x) by A5, A10, VALUED_1:def_1
.= 1 + (f2 . x) by A3, A10
.= 1 + (x #Z (1 + 1)) by A2, TAYLOR_1:def_1
.= 1 + ((x #Z 1) * (x #Z 1)) by TAYLOR_1:1
.= 1 + (x * (x #Z 1)) by PREPOWER:35
.= 1 + (x * x) by PREPOWER:35 ;
then  (f1 + f2) . x > 0 by XREAL_1:34, XREAL_1:63;
then  diff ((ln * (f1 + f2)),x) = (diff ((f1 + f2),x)) / ((f1 + f2) . x) by A11, TAYLOR_1:20
.= (((f1 + f2) `| Z) . x) / ((f1 + f2) . x) by A6, A10, FDIFF_1:def_7
.= (2 * x) / (1 + (x ^2)) by A2, A3, A5, A10, A12, Th101 ;
hence (((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = (1 / 2) * ((2 * x) / (1 + (x ^2))) by A1, A9, A10, FDIFF_1:20
.= x / (1 + (x ^2)) ;
::_thesis: verum
end;
hence  ( (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) ) ) by A1, A9, FDIFF_1:20; ::_thesis: verum
end;

theorem :: SIN_COS9:103
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( ((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x ) )
proof
let Z be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( ((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x ) )
let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) implies ( ((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x ) ) )
assume that
A1: Z c= dom (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) and
A2: Z c= ].(- 1),1.[ and
A3: f2 = #Z 2 and
A4: for x being Real st x in Z holds
f1 . x = 1 ; ::_thesis: ( ((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x ) )
 Z c= (dom ((id Z) (#) arctan)) /\ (dom ((1 / 2) (#) (ln * (f1 + f2)))) by A1, VALUED_1:12;
then A5: Z c= dom ((1 / 2) (#) (ln * (f1 + f2))) by XBOOLE_1:18;
then A6: (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z by A3, A4, Th102;
A7: (id Z) (#) arctan is_differentiable_on Z by A2, Th95;
 for x being Real st x in Z holds
((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x
proof
let x be Real; ::_thesis: ( x in Z implies ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x )
assume A8: x in Z ; ::_thesis: ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x
hence ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = (diff (((id Z) (#) arctan),x)) - (diff (((1 / 2) (#) (ln * (f1 + f2))),x)) by A1, A7, A6, FDIFF_1:19
.= ((((id Z) (#) arctan) `| Z) . x) - (diff (((1 / 2) (#) (ln * (f1 + f2))),x)) by A7, A8, FDIFF_1:def_7
.= ((((id Z) (#) arctan) `| Z) . x) - ((((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A6, A8, FDIFF_1:def_7
.= ((arctan . x) + (x / (1 + (x ^2)))) - ((((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A2, A8, Th95
.= ((arctan . x) + (x / (1 + (x ^2)))) - (x / (1 + (x ^2))) by A3, A4, A5, A8, Th102
.= arctan . x ;
::_thesis: verum
end;
hence  ( ((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x ) ) by A1, A7, A6, FDIFF_1:19; ::_thesis: verum
end;

theorem :: SIN_COS9:104
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( ((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x ) )
proof
let Z be open Subset of REAL; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( ((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x ) )
let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) implies ( ((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x ) ) )
assume that
A1: Z c= dom (((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) and
A2: Z c= ].(- 1),1.[ and
A3: f2 = #Z 2 and
A4: for x being Real st x in Z holds
f1 . x = 1 ; ::_thesis: ( ((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x ) )
 Z c= (dom ((id Z) (#) arccot)) /\ (dom ((1 / 2) (#) (ln * (f1 + f2)))) by A1, VALUED_1:def_1;
then A5: Z c= dom ((1 / 2) (#) (ln * (f1 + f2))) by XBOOLE_1:18;
then A6: (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z by A3, A4, Th102;
A7: (id Z) (#) arccot is_differentiable_on Z by A2, Th96;
 for x being Real st x in Z holds
((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x
proof
let x be Real; ::_thesis: ( x in Z implies ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x )
assume A8: x in Z ; ::_thesis: ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x
hence ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = (diff (((id Z) (#) arccot),x)) + (diff (((1 / 2) (#) (ln * (f1 + f2))),x)) by A1, A7, A6, FDIFF_1:18
.= ((((id Z) (#) arccot) `| Z) . x) + (diff (((1 / 2) (#) (ln * (f1 + f2))),x)) by A7, A8, FDIFF_1:def_7
.= ((((id Z) (#) arccot) `| Z) . x) + ((((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A6, A8, FDIFF_1:def_7
.= ((arccot . x) - (x / (1 + (x ^2)))) + ((((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A2, A8, Th96
.= ((arccot . x) - (x / (1 + (x ^2)))) + (x / (1 + (x ^2))) by A3, A4, A5, A8, Th102
.= arccot . x ;
::_thesis: verum
end;
hence  ( ((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x ) ) by A1, A7, A6, FDIFF_1:18; ::_thesis: verum
end;

theorem Th105: :: SIN_COS9:105
for r being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((id Z) (#) (arctan * f)) & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) holds
( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) )
proof
let r be Real; ::_thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((id Z) (#) (arctan * f)) & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) holds
( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) )
let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((id Z) (#) (arctan * f)) & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) holds
( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) )
let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((id Z) (#) (arctan * f)) & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) implies ( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) ) )
assume that
A1: Z c= dom ((id Z) (#) (arctan * f)) and
A2: for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ; ::_thesis: ( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) )
A3: Z c= (dom (id Z)) /\ (dom (arctan * f)) by A1, VALUED_1:def_4;
then A4: Z c= dom (id Z) by XBOOLE_1:18;
A5: Z c= dom (arctan * f) by A3, XBOOLE_1:18;
 for x being Real st x in Z holds
f . x = ((1 / r) * x) + 0
proof
let x be Real; ::_thesis: ( x in Z implies f . x = ((1 / r) * x) + 0 )
assume  x in Z ; ::_thesis: f . x = ((1 / r) * x) + 0
then  f . x = x / r by A2;
hence  f . x = ((1 / r) * x) + 0 ; ::_thesis: verum
end;
then A6: for x being Real st x in Z holds
( f . x = ((1 / r) * x) + 0 & f . x > - 1 & f . x < 1 ) by A2;
then A7: arctan * f is_differentiable_on Z by A5, Th87;
A8: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
then A9: id Z is_differentiable_on Z by A4, FDIFF_1:23;
A10: for x being Real st x in Z holds
((arctan * f) `| Z) . x = 1 / (r * (1 + ((x / r) ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((arctan * f) `| Z) . x = 1 / (r * (1 + ((x / r) ^2))) )
assume  x in Z ; ::_thesis: ((arctan * f) `| Z) . x = 1 / (r * (1 + ((x / r) ^2)))
then ((arctan * f) `| Z) . x = (1 / r) / (1 + ((((1 / r) * x) + 0) ^2)) by A6, A5, Th87
.= 1 / (r * (1 + ((x / r) ^2))) by XCMPLX_1:78 ;
hence  ((arctan * f) `| Z) . x = 1 / (r * (1 + ((x / r) ^2))) ; ::_thesis: verum
end;
 for x being Real st x in Z holds
(((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies (((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) )
assume A11: x in Z ; ::_thesis: (((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2))))
then A12: (arctan * f) . x = arctan . (f . x) by A5, FUNCT_1:12
.= arctan . (x / r) by A2, A11 ;
(((id Z) (#) (arctan * f)) `| Z) . x = (((arctan * f) . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff ((arctan * f),x))) by A1, A9, A7, A11, FDIFF_1:21
.= (((arctan * f) . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff ((arctan * f),x))) by A9, A11, FDIFF_1:def_7
.= (((arctan * f) . x) * 1) + (((id Z) . x) * (diff ((arctan * f),x))) by A4, A8, A11, FDIFF_1:23
.= (((arctan * f) . x) * 1) + (((id Z) . x) * (((arctan * f) `| Z) . x)) by A7, A11, FDIFF_1:def_7
.= ((arctan * f) . x) + (x * (((arctan * f) `| Z) . x)) by A11, FUNCT_1:18
.= (arctan . (x / r)) + (x * (1 / (r * (1 + ((x / r) ^2))))) by A10, A11, A12
.= (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ;
hence  (((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ; ::_thesis: verum
end;
hence  ( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) ) by A1, A9, A7, FDIFF_1:21; ::_thesis: verum
end;

theorem Th106: :: SIN_COS9:106
for r being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((id Z) (#) (arccot * f)) & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) holds
( (id Z) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) ) )
proof
let r be Real; ::_thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((id Z) (#) (arccot * f)) & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) holds
( (id Z) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) ) )
let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom ((id Z) (#) (arccot * f)) & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) holds
( (id Z) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) ) )
let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((id Z) (#) (arccot * f)) & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) implies ( (id Z) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) ) ) )
assume that
A1: Z c= dom ((id Z) (#) (arccot * f)) and
A2: for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ; ::_thesis: ( (id Z) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) ) )
A3: Z c= (dom (id Z)) /\ (dom (arccot * f)) by A1, VALUED_1:def_4;
then A4: Z c= dom (id Z) by XBOOLE_1:18;
A5: Z c= dom (arccot * f) by A3, XBOOLE_1:18;
 for x being Real st x in Z holds
f . x = ((1 / r) * x) + 0
proof
let x be Real; ::_thesis: ( x in Z implies f . x = ((1 / r) * x) + 0 )
assume  x in Z ; ::_thesis: f . x = ((1 / r) * x) + 0
then  f . x = x / r by A2;
hence  f . x = ((1 / r) * x) + 0 ; ::_thesis: verum
end;
then A6: for x being Real st x in Z holds
( f . x = ((1 / r) * x) + 0 & f . x > - 1 & f . x < 1 ) by A2;
then A7: arccot * f is_differentiable_on Z by A5, Th88;
A8: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
then A9: id Z is_differentiable_on Z by A4, FDIFF_1:23;
A10: for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (1 / (r * (1 + ((x / r) ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies ((arccot * f) `| Z) . x = - (1 / (r * (1 + ((x / r) ^2)))) )
assume  x in Z ; ::_thesis: ((arccot * f) `| Z) . x = - (1 / (r * (1 + ((x / r) ^2))))
then ((arccot * f) `| Z) . x = - ((1 / r) / (1 + ((((1 / r) * x) + 0) ^2))) by A6, A5, Th88
.= - (1 / (r * (1 + ((x / r) ^2)))) by XCMPLX_1:78 ;
hence  ((arccot * f) `| Z) . x = - (1 / (r * (1 + ((x / r) ^2)))) ; ::_thesis: verum
end;
 for x being Real st x in Z holds
(((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies (((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) )
assume A11: x in Z ; ::_thesis: (((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2))))
then A12: (arccot * f) . x = arccot . (f . x) by A5, FUNCT_1:12
.= arccot . (x / r) by A2, A11 ;
(((id Z) (#) (arccot * f)) `| Z) . x = (((arccot * f) . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff ((arccot * f),x))) by A1, A9, A7, A11, FDIFF_1:21
.= (((arccot * f) . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff ((arccot * f),x))) by A9, A11, FDIFF_1:def_7
.= (((arccot * f) . x) * 1) + (((id Z) . x) * (diff ((arccot * f),x))) by A4, A8, A11, FDIFF_1:23
.= (((arccot * f) . x) * 1) + (((id Z) . x) * (((arccot * f) `| Z) . x)) by A7, A11, FDIFF_1:def_7
.= ((arccot * f) . x) + (x * (((arccot * f) `| Z) . x)) by A11, FUNCT_1:18
.= (arccot . (x / r)) + (x * (- (1 / (r * (1 + ((x / r) ^2)))))) by A10, A11, A12
.= (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) ;
hence  (((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) ; ::_thesis: verum
end;
hence  ( (id Z) (#) (arccot * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arccot * f)) `| Z) . x = (arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2)))) ) ) by A1, A9, A7, FDIFF_1:21; ::_thesis: verum
end;

theorem Th107: :: SIN_COS9:107
for r being Real
for Z being open Subset of REAL
for f1, f2, f being PartFunc of REAL,REAL st Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) )
proof
let r be Real; ::_thesis: for Z being open Subset of REAL
for f1, f2, f being PartFunc of REAL,REAL st Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) )
let Z be open Subset of REAL; ::_thesis: for f1, f2, f being PartFunc of REAL,REAL st Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) )
let f1, f2, f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) implies ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) ) )
assume that
A1: Z c= dom (f1 + f2) and
A2: for x being Real st x in Z holds
f1 . x = 1 and
A3: f2 = (#Z 2) * f and
A4: for x being Real st x in Z holds
f . x = x / r ; ::_thesis: ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) )
A5: for x being Real st x in Z holds
f1 . x = (0 * x) + 1 by A2;
A6: Z c= (dom f1) /\ (dom f2) by A1, VALUED_1:def_1;
then A7: Z c= dom f1 by XBOOLE_1:18;
then A8: f1 is_differentiable_on Z by A5, FDIFF_1:23;
A9: for x being Real st x in Z holds
f . x = ((1 / r) * x) + 0
proof
let x be Real; ::_thesis: ( x in Z implies f . x = ((1 / r) * x) + 0 )
assume  x in Z ; ::_thesis: f . x = ((1 / r) * x) + 0
hence f . x = x / r by A4
.= ((1 / r) * x) + 0 ;
::_thesis: verum
end;
A10: for x being Real st x in Z holds
f2 is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies f2 is_differentiable_in x )
 Z c= dom ((#Z 2) * f) by A3, A6, XBOOLE_1:18;
then  for y being set st y in Z holds
y in dom f by FUNCT_1:11;
then  Z c= dom f by TARSKI:def_3;
then A11: f is_differentiable_on Z by A9, FDIFF_1:23;
assume  x in Z ; ::_thesis: f2 is_differentiable_in x
then  f is_differentiable_in x by A11, FDIFF_1:9;
hence  f2 is_differentiable_in x by A3, TAYLOR_1:3; ::_thesis: verum
end;
 Z c= dom f2 by A6, XBOOLE_1:18;
then A12: f2 is_differentiable_on Z by A10, FDIFF_1:9;
A13: for x being Real st x in Z holds
(f2 `| Z) . x = (2 * x) / (r ^2)
proof
let x be Real; ::_thesis: ( x in Z implies (f2 `| Z) . x = (2 * x) / (r ^2) )
assume A14: x in Z ; ::_thesis: (f2 `| Z) . x = (2 * x) / (r ^2)
 Z c= dom ((#Z 2) * f) by A3, A6, XBOOLE_1:18;
then  for y being set st y in Z holds
y in dom f by FUNCT_1:11;
then A15: Z c= dom f by TARSKI:def_3;
then A16: f is_differentiable_on Z by A9, FDIFF_1:23;
then A17: f is_differentiable_in x by A14, FDIFF_1:9;
(f2 `| Z) . x = diff (((#Z 2) * f),x) by A3, A12, A14, FDIFF_1:def_7
.= (2 * ((f . x) #Z (2 - 1))) * (diff (f,x)) by A17, TAYLOR_1:3
.= (2 * (f . x)) * (diff (f,x)) by PREPOWER:35
.= (2 * (x / r)) * (diff (f,x)) by A4, A14
.= (2 * (x / r)) * ((f `| Z) . x) by A14, A16, FDIFF_1:def_7
.= (2 * (x / r)) * (1 / r) by A9, A14, A15, FDIFF_1:23
.= 2 * ((x / r) * (1 / r))
.= 2 * ((x * 1) / (r * r)) by XCMPLX_1:76
.= (2 * x) / (r ^2) ;
hence  (f2 `| Z) . x = (2 * x) / (r ^2) ; ::_thesis: verum
end;
 for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2)
proof
let x be Real; ::_thesis: ( x in Z implies ((f1 + f2) `| Z) . x = (2 * x) / (r ^2) )
assume A18: x in Z ; ::_thesis: ((f1 + f2) `| Z) . x = (2 * x) / (r ^2)
then ((f1 + f2) `| Z) . x = (diff (f1,x)) + (diff (f2,x)) by A1, A8, A12, FDIFF_1:18
.= ((f1 `| Z) . x) + (diff (f2,x)) by A8, A18, FDIFF_1:def_7
.= ((f1 `| Z) . x) + ((f2 `| Z) . x) by A12, A18, FDIFF_1:def_7
.= 0 + ((f2 `| Z) . x) by A7, A5, A18, FDIFF_1:23
.= (2 * x) / (r ^2) by A13, A18 ;
hence  ((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ; ::_thesis: verum
end;
hence  ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) ) by A1, A8, A12, FDIFF_1:18; ::_thesis: verum
end;

theorem Th108: :: SIN_COS9:108
for r being Real
for Z being open Subset of REAL
for f1, f2, f being PartFunc of REAL,REAL st Z c= dom ((r / 2) (#) (ln * (f1 + f2))) & ( for x being Real st x in Z holds
f1 . x = 1 ) & r <> 0 & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2))) ) )
proof
let r be Real; ::_thesis: for Z being open Subset of REAL
for f1, f2, f being PartFunc of REAL,REAL st Z c= dom ((r / 2) (#) (ln * (f1 + f2))) & ( for x being Real st x in Z holds
f1 . x = 1 ) & r <> 0 & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2))) ) )
let Z be open Subset of REAL; ::_thesis: for f1, f2, f being PartFunc of REAL,REAL st Z c= dom ((r / 2) (#) (ln * (f1 + f2))) & ( for x being Real st x in Z holds
f1 . x = 1 ) & r <> 0 & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2))) ) )
let f1, f2, f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom ((r / 2) (#) (ln * (f1 + f2))) & ( for x being Real st x in Z holds
f1 . x = 1 ) & r <> 0 & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) implies ( (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2))) ) ) )
assume that
A1: Z c= dom ((r / 2) (#) (ln * (f1 + f2))) and
A2: for x being Real st x in Z holds
f1 . x = 1 and
A3: r <> 0 and
A4: f2 = (#Z 2) * f and
A5: for x being Real st x in Z holds
f . x = x / r ; ::_thesis: ( (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2))) ) )
A6: Z c= dom (ln * (f1 + f2)) by A1, VALUED_1:def_5;
then  for y being set st y in Z holds
y in dom (f1 + f2) by FUNCT_1:11;
then A7: Z c= dom (f1 + f2) by TARSKI:def_3;
then A8: f1 + f2 is_differentiable_on Z by A2, A4, A5, Th107;
 dom (f1 + f2) = (dom f1) /\ (dom f2) by VALUED_1:def_1;
then A9: Z c= dom f2 by A7, XBOOLE_1:18;
 for x being Real st x in Z holds
ln * (f1 + f2) is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies ln * (f1 + f2) is_differentiable_in x )
set g = #Z 2;
assume A10: x in Z ; ::_thesis: ln * (f1 + f2) is_differentiable_in x
then (f1 + f2) . x = (f1 . x) + (f2 . x) by A7, VALUED_1:def_1
.= 1 + (((#Z 2) * f) . x) by A2, A4, A10
.= 1 + ((#Z 2) . (f . x)) by A4, A9, A10, FUNCT_1:12
.= 1 + ((#Z 2) . (x / r)) by A5, A10
.= 1 + ((x / r) #Z (1 + 1)) by TAYLOR_1:def_1
.= 1 + (((x / r) #Z 1) * ((x / r) #Z 1)) by TAYLOR_1:1
.= 1 + ((x / r) * ((x / r) #Z 1)) by PREPOWER:35
.= 1 + ((x / r) * (x / r)) by PREPOWER:35 ;
then A11: (f1 + f2) . x > 0 by XREAL_1:34, XREAL_1:63;
 f1 + f2 is_differentiable_in x by A8, A10, FDIFF_1:9;
hence  ln * (f1 + f2) is_differentiable_in x by A11, TAYLOR_1:20; ::_thesis: verum
end;
then A12: ln * (f1 + f2) is_differentiable_on Z by A6, FDIFF_1:9;
 for x being Real st x in Z holds
(((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies (((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2))) )
set g = #Z 2;
assume A13: x in Z ; ::_thesis: (((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2)))
then A14: f1 + f2 is_differentiable_in x by A8, FDIFF_1:9;
A15: (f1 + f2) . x = (f1 . x) + (f2 . x) by A7, A13, VALUED_1:def_1
.= 1 + (((#Z 2) * f) . x) by A2, A4, A13
.= 1 + ((#Z 2) . (f . x)) by A4, A9, A13, FUNCT_1:12
.= 1 + ((#Z 2) . (x / r)) by A5, A13
.= 1 + ((x / r) #Z (1 + 1)) by TAYLOR_1:def_1
.= 1 + (((x / r) #Z 1) * ((x / r) #Z 1)) by TAYLOR_1:1
.= 1 + ((x / r) * ((x / r) #Z 1)) by PREPOWER:35
.= 1 + ((x / r) * (x / r)) by PREPOWER:35 ;
then  (f1 + f2) . x > 0 by XREAL_1:34, XREAL_1:63;
then A16: diff ((ln * (f1 + f2)),x) = (diff ((f1 + f2),x)) / ((f1 + f2) . x) by A14, TAYLOR_1:20
.= (((f1 + f2) `| Z) . x) / ((f1 + f2) . x) by A8, A13, FDIFF_1:def_7
.= ((2 * x) / (r ^2)) / (1 + ((x / r) ^2)) by A2, A4, A5, A7, A13, A15, Th107 ;
thus (((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = (r / 2) * (diff ((ln * (f1 + f2)),x)) by A1, A12, A13, FDIFF_1:20
.= ((r * x) / (r ^2)) / (1 + ((x / r) ^2)) by A16
.= ((r / r) * (x / r)) / (1 + ((x / r) ^2)) by XCMPLX_1:76
.= (1 * (x / r)) / (1 + ((x / r) ^2)) by A3, XCMPLX_1:60
.= x / (r * (1 + ((x / r) ^2))) by XCMPLX_1:78 ; ::_thesis: verum
end;
hence  ( (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2))) ) ) by A1, A12, FDIFF_1:20; ::_thesis: verum
end;

theorem :: SIN_COS9:109
for r being Real
for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) )
proof
let r be Real; ::_thesis: for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) )
let Z be open Subset of REAL; ::_thesis: for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) )
let f, f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) implies ( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) ) )
assume that
A1: Z c= dom (((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) and
A2: r <> 0 and
A3: for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) and
A4: for x being Real st x in Z holds
f1 . x = 1 and
A5: f2 = (#Z 2) * f and
A6: for x being Real st x in Z holds
f . x = x / r ; ::_thesis: ( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) )
A7: Z c= (dom ((id Z) (#) (arctan * f))) /\ (dom ((r / 2) (#) (ln * (f1 + f2)))) by A1, VALUED_1:12;
then A8: Z c= dom ((r / 2) (#) (ln * (f1 + f2))) by XBOOLE_1:18;
then A9: (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z by A2, A4, A5, A6, Th108;
A10: Z c= dom ((id Z) (#) (arctan * f)) by A7, XBOOLE_1:18;
then A11: (id Z) (#) (arctan * f) is_differentiable_on Z by A3, Th105;
 for x being Real st x in Z holds
((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r)
proof
let x be Real; ::_thesis: ( x in Z implies ((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) )
assume A12: x in Z ; ::_thesis: ((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r)
hence ((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = (diff (((id Z) (#) (arctan * f)),x)) - (diff (((r / 2) (#) (ln * (f1 + f2))),x)) by A1, A11, A9, FDIFF_1:19
.= ((((id Z) (#) (arctan * f)) `| Z) . x) - (diff (((r / 2) (#) (ln * (f1 + f2))),x)) by A11, A12, FDIFF_1:def_7
.= ((((id Z) (#) (arctan * f)) `| Z) . x) - ((((r / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A9, A12, FDIFF_1:def_7
.= ((arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2))))) - ((((r / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A3, A10, A12, Th105
.= ((arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2))))) - (x / (r * (1 + ((x / r) ^2)))) by A2, A4, A5, A6, A8, A12, Th108
.= arctan . (x / r) ;
::_thesis: verum
end;
hence  ( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) ) by A1, A11, A9, FDIFF_1:19; ::_thesis: verum
end;

theorem :: SIN_COS9:110
for r being Real
for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) )
proof
let r be Real; ::_thesis: for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) )
let Z be open Subset of REAL; ::_thesis: for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) )
let f, f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) implies ( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) ) )
assume that
A1: Z c= dom (((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) and
A2: r <> 0 and
A3: for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) and
A4: for x being Real st x in Z holds
f1 . x = 1 and
A5: f2 = (#Z 2) * f and
A6: for x being Real st x in Z holds
f . x = x / r ; ::_thesis: ( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) )
A7: Z c= (dom ((id Z) (#) (arccot * f))) /\ (dom ((r / 2) (#) (ln * (f1 + f2)))) by A1, VALUED_1:def_1;
then A8: Z c= dom ((r / 2) (#) (ln * (f1 + f2))) by XBOOLE_1:18;
then A9: (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z by A2, A4, A5, A6, Th108;
A10: Z c= dom ((id Z) (#) (arccot * f)) by A7, XBOOLE_1:18;
then A11: (id Z) (#) (arccot * f) is_differentiable_on Z by A3, Th106;
 for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r)
proof
let x be Real; ::_thesis: ( x in Z implies ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) )
assume A12: x in Z ; ::_thesis: ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r)
hence ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = (diff (((id Z) (#) (arccot * f)),x)) + (diff (((r / 2) (#) (ln * (f1 + f2))),x)) by A1, A11, A9, FDIFF_1:18
.= ((((id Z) (#) (arccot * f)) `| Z) . x) + (diff (((r / 2) (#) (ln * (f1 + f2))),x)) by A11, A12, FDIFF_1:def_7
.= ((((id Z) (#) (arccot * f)) `| Z) . x) + ((((r / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A9, A12, FDIFF_1:def_7
.= ((arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2))))) + ((((r / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A3, A10, A12, Th106
.= ((arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2))))) + (x / (r * (1 + ((x / r) ^2)))) by A2, A4, A5, A6, A8, A12, Th108
.= arccot . (x / r) ;
::_thesis: verum
end;
hence  ( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) ) by A1, A11, A9, FDIFF_1:18; ::_thesis: verum
end;

theorem :: SIN_COS9:111
for Z being open Subset of REAL st not 0 in Z & Z c= dom (arctan * ((id Z) ^)) & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds
( arctan * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * ((id Z) ^)) `| Z) . x = - (1 / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( not 0 in Z & Z c= dom (arctan * ((id Z) ^)) & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) implies ( arctan * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * ((id Z) ^)) `| Z) . x = - (1 / (1 + (x ^2))) ) ) )
set f = id Z;
assume that
A1: not 0 in Z and
A2: Z c= dom (arctan * ((id Z) ^)) and
A3: for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ; ::_thesis: ( arctan * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * ((id Z) ^)) `| Z) . x = - (1 / (1 + (x ^2))) ) )
 dom (arctan * ((id Z) ^)) c= dom ((id Z) ^) by RELAT_1:25;
then A4: Z c= dom ((id Z) ^) by A2, XBOOLE_1:1;
A5: (id Z) ^ is_differentiable_on Z by A1, FDIFF_5:4;
A6: for x being Real st x in Z holds
arctan * ((id Z) ^) is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arctan * ((id Z) ^) is_differentiable_in x )
assume A7: x in Z ; ::_thesis: arctan * ((id Z) ^) is_differentiable_in x
then A8: ((id Z) ^) . x > - 1 by A3;
A9: ((id Z) ^) . x < 1 by A3, A7;
 (id Z) ^ is_differentiable_in x by A5, A7, FDIFF_1:9;
hence  arctan * ((id Z) ^) is_differentiable_in x by A8, A9, Th85; ::_thesis: verum
end;
then A10: arctan * ((id Z) ^) is_differentiable_on Z by A2, FDIFF_1:9;
 for x being Real st x in Z holds
((arctan * ((id Z) ^)) `| Z) . x = - (1 / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((arctan * ((id Z) ^)) `| Z) . x = - (1 / (1 + (x ^2))) )
assume A11: x in Z ; ::_thesis: ((arctan * ((id Z) ^)) `| Z) . x = - (1 / (1 + (x ^2)))
then A12: (id Z) ^ is_differentiable_in x by A5, FDIFF_1:9;
A13: ((id Z) ^) . x < 1 by A3, A11;
A14: ((id Z) ^) . x > - 1 by A3, A11;
 (id Z) . x = x by A11, FUNCT_1:18;
then  x <> 0 by A4, A11, RFUNCT_1:3;
then A15: x ^2 <> 0 by SQUARE_1:12;
((arctan * ((id Z) ^)) `| Z) . x = diff ((arctan * ((id Z) ^)),x) by A10, A11, FDIFF_1:def_7
.= (diff (((id Z) ^),x)) / (1 + ((((id Z) ^) . x) ^2)) by A12, A14, A13, Th85
.= ((((id Z) ^) `| Z) . x) / (1 + ((((id Z) ^) . x) ^2)) by A5, A11, FDIFF_1:def_7
.= (- (1 / (x ^2))) / (1 + ((((id Z) ^) . x) ^2)) by A1, A11, FDIFF_5:4
.= (- (1 / (x ^2))) / (1 + ((((id Z) . x) ") ^2)) by A4, A11, RFUNCT_1:def_2
.= (- (1 / (x ^2))) / (1 + ((1 / x) ^2)) by A11, FUNCT_1:18
.= - ((1 / (x ^2)) / (1 + ((1 / x) ^2)))
.= - (1 / ((x ^2) * (1 + ((1 / x) ^2)))) by XCMPLX_1:78
.= - (1 / (((x ^2) * 1) + ((x ^2) * ((1 / x) ^2))))
.= - (1 / ((x ^2) + ((x ^2) * (1 / (x * x))))) by XCMPLX_1:102
.= - (1 / ((x ^2) + (((x ^2) * 1) / (x ^2))))
.= - (1 / (1 + (x ^2))) by A15, XCMPLX_1:60 ;
hence  ((arctan * ((id Z) ^)) `| Z) . x = - (1 / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( arctan * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * ((id Z) ^)) `| Z) . x = - (1 / (1 + (x ^2))) ) ) by A2, A6, FDIFF_1:9; ::_thesis: verum
end;

theorem :: SIN_COS9:112
for Z being open Subset of REAL st not 0 in Z & Z c= dom (arccot * ((id Z) ^)) & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds
( arccot * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * ((id Z) ^)) `| Z) . x = 1 / (1 + (x ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( not 0 in Z & Z c= dom (arccot * ((id Z) ^)) & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) implies ( arccot * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * ((id Z) ^)) `| Z) . x = 1 / (1 + (x ^2)) ) ) )
set f = id Z;
assume that
A1: not 0 in Z and
A2: Z c= dom (arccot * ((id Z) ^)) and
A3: for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ; ::_thesis: ( arccot * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * ((id Z) ^)) `| Z) . x = 1 / (1 + (x ^2)) ) )
 dom (arccot * ((id Z) ^)) c= dom ((id Z) ^) by RELAT_1:25;
then A4: Z c= dom ((id Z) ^) by A2, XBOOLE_1:1;
A5: (id Z) ^ is_differentiable_on Z by A1, FDIFF_5:4;
A6: for x being Real st x in Z holds
arccot * ((id Z) ^) is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arccot * ((id Z) ^) is_differentiable_in x )
assume A7: x in Z ; ::_thesis: arccot * ((id Z) ^) is_differentiable_in x
then A8: ((id Z) ^) . x > - 1 by A3;
A9: ((id Z) ^) . x < 1 by A3, A7;
 (id Z) ^ is_differentiable_in x by A5, A7, FDIFF_1:9;
hence  arccot * ((id Z) ^) is_differentiable_in x by A8, A9, Th86; ::_thesis: verum
end;
then A10: arccot * ((id Z) ^) is_differentiable_on Z by A2, FDIFF_1:9;
 for x being Real st x in Z holds
((arccot * ((id Z) ^)) `| Z) . x = 1 / (1 + (x ^2))
proof
let x be Real; ::_thesis: ( x in Z implies ((arccot * ((id Z) ^)) `| Z) . x = 1 / (1 + (x ^2)) )
assume A11: x in Z ; ::_thesis: ((arccot * ((id Z) ^)) `| Z) . x = 1 / (1 + (x ^2))
then A12: (id Z) ^ is_differentiable_in x by A5, FDIFF_1:9;
A13: ((id Z) ^) . x < 1 by A3, A11;
A14: ((id Z) ^) . x > - 1 by A3, A11;
 (id Z) . x = x by A11, FUNCT_1:18;
then  x <> 0 by A4, A11, RFUNCT_1:3;
then A15: x ^2 <> 0 by SQUARE_1:12;
((arccot * ((id Z) ^)) `| Z) . x = diff ((arccot * ((id Z) ^)),x) by A10, A11, FDIFF_1:def_7
.= - ((diff (((id Z) ^),x)) / (1 + ((((id Z) ^) . x) ^2))) by A12, A14, A13, Th86
.= - (((((id Z) ^) `| Z) . x) / (1 + ((((id Z) ^) . x) ^2))) by A5, A11, FDIFF_1:def_7
.= - ((- (1 / (x ^2))) / (1 + ((((id Z) ^) . x) ^2))) by A1, A11, FDIFF_5:4
.= - ((- (1 / (x ^2))) / (1 + ((((id Z) . x) ") ^2))) by A4, A11, RFUNCT_1:def_2
.= - ((- (1 / (x ^2))) / (1 + ((1 / x) ^2))) by A11, FUNCT_1:18
.= (1 / (x ^2)) / (1 + ((1 / x) ^2))
.= 1 / ((x ^2) * (1 + ((1 / x) ^2))) by XCMPLX_1:78
.= 1 / (((x ^2) * 1) + ((x ^2) * ((1 / x) ^2)))
.= 1 / ((x ^2) + ((x ^2) * (1 / (x * x)))) by XCMPLX_1:102
.= 1 / ((x ^2) + (((x ^2) * 1) / (x ^2)))
.= 1 / (1 + (x ^2)) by A15, XCMPLX_1:60 ;
hence  ((arccot * ((id Z) ^)) `| Z) . x = 1 / (1 + (x ^2)) ; ::_thesis: verum
end;
hence  ( arccot * ((id Z) ^) is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * ((id Z) ^)) `| Z) . x = 1 / (1 + (x ^2)) ) ) by A2, A6, FDIFF_1:9; ::_thesis: verum
end;

theorem :: SIN_COS9:113
for h, r, s being Real
for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (arctan * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds
( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = r + (s * x) ) & f2 = #Z 2 holds
( arctan * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) )
proof
let h, r, s be Real; ::_thesis: for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (arctan * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds
( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = r + (s * x) ) & f2 = #Z 2 holds
( arctan * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) )
let Z be open Subset of REAL; ::_thesis: for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (arctan * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds
( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = r + (s * x) ) & f2 = #Z 2 holds
( arctan * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) )
let f, f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (arctan * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds
( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = r + (s * x) ) & f2 = #Z 2 implies ( arctan * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) ) )
assume that
A1: Z c= dom (arctan * f) and
A2: f = f1 + (h (#) f2) and
A3: for x being Real st x in Z holds
( f . x > - 1 & f . x < 1 ) and
A4: for x being Real st x in Z holds
f1 . x = r + (s * x) and
A5: f2 = #Z 2 ; ::_thesis: ( arctan * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) )
 dom (arctan * f) c= dom f by RELAT_1:25;
then A6: Z c= dom (f1 + (h (#) f2)) by A1, A2, XBOOLE_1:1;
then  Z c= (dom f1) /\ (dom (h (#) f2)) by VALUED_1:def_1;
then A7: Z c= dom (h (#) f2) by XBOOLE_1:18;
A8: f1 + (h (#) f2) is_differentiable_on Z by A4, A5, A6, FDIFF_4:12;
A9: for x being Real st x in Z holds
arctan * (f1 + (h (#) f2)) is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arctan * (f1 + (h (#) f2)) is_differentiable_in x )
assume A10: x in Z ; ::_thesis: arctan * (f1 + (h (#) f2)) is_differentiable_in x
then A11: f . x > - 1 by A3;
A12: f . x < 1 by A3, A10;
 f is_differentiable_in x by A2, A8, A10, FDIFF_1:9;
hence  arctan * (f1 + (h (#) f2)) is_differentiable_in x by A2, A11, A12, Th85; ::_thesis: verum
end;
then A13: arctan * (f1 + (h (#) f2)) is_differentiable_on Z by A1, A2, FDIFF_1:9;
 for x being Real st x in Z holds
((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))
proof
let x be Real; ::_thesis: ( x in Z implies ((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) )
assume A14: x in Z ; ::_thesis: ((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))
then A15: (f1 + (h (#) f2)) . x = (f1 . x) + ((h (#) f2) . x) by A6, VALUED_1:def_1
.= (f1 . x) + (h * (f2 . x)) by A7, A14, VALUED_1:def_5
.= (r + (s * x)) + (h * (f2 . x)) by A4, A14
.= (r + (s * x)) + (h * (x #Z (1 + 1))) by A5, TAYLOR_1:def_1
.= (r + (s * x)) + (h * ((x #Z 1) * (x #Z 1))) by TAYLOR_1:1
.= (r + (s * x)) + (h * (x * (x #Z 1))) by PREPOWER:35
.= (r + (s * x)) + (h * (x ^2)) by PREPOWER:35 ;
A16: f is_differentiable_in x by A2, A8, A14, FDIFF_1:9;
A17: f . x > - 1 by A3, A14;
A18: f . x < 1 by A3, A14;
((arctan * (f1 + (h (#) f2))) `| Z) . x = diff ((arctan * f),x) by A2, A13, A14, FDIFF_1:def_7
.= (diff (f,x)) / (1 + ((f . x) ^2)) by A16, A17, A18, Th85
.= (((f1 + (h (#) f2)) `| Z) . x) / (1 + ((f . x) ^2)) by A2, A8, A14, FDIFF_1:def_7
.= (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) by A2, A4, A5, A6, A14, A15, FDIFF_4:12 ;
hence  ((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ; ::_thesis: verum
end;
hence  ( arctan * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) ) by A1, A2, A9, FDIFF_1:9; ::_thesis: verum
end;

theorem :: SIN_COS9:114
for h, r, s being Real
for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (arccot * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds
( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = r + (s * x) ) & f2 = #Z 2 holds
( arccot * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) ) )
proof
let h, r, s be Real; ::_thesis: for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (arccot * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds
( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = r + (s * x) ) & f2 = #Z 2 holds
( arccot * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) ) )
let Z be open Subset of REAL; ::_thesis: for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (arccot * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds
( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = r + (s * x) ) & f2 = #Z 2 holds
( arccot * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) ) )
let f, f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (arccot * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds
( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = r + (s * x) ) & f2 = #Z 2 implies ( arccot * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) ) ) )
assume that
A1: Z c= dom (arccot * f) and
A2: f = f1 + (h (#) f2) and
A3: for x being Real st x in Z holds
( f . x > - 1 & f . x < 1 ) and
A4: for x being Real st x in Z holds
f1 . x = r + (s * x) and
A5: f2 = #Z 2 ; ::_thesis: ( arccot * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) ) )
 dom (arccot * f) c= dom f by RELAT_1:25;
then A6: Z c= dom (f1 + (h (#) f2)) by A1, A2, XBOOLE_1:1;
then  Z c= (dom f1) /\ (dom (h (#) f2)) by VALUED_1:def_1;
then A7: Z c= dom (h (#) f2) by XBOOLE_1:18;
A8: f1 + (h (#) f2) is_differentiable_on Z by A4, A5, A6, FDIFF_4:12;
A9: for x being Real st x in Z holds
arccot * (f1 + (h (#) f2)) is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arccot * (f1 + (h (#) f2)) is_differentiable_in x )
assume A10: x in Z ; ::_thesis: arccot * (f1 + (h (#) f2)) is_differentiable_in x
then A11: f . x > - 1 by A3;
A12: f . x < 1 by A3, A10;
 f is_differentiable_in x by A2, A8, A10, FDIFF_1:9;
hence  arccot * (f1 + (h (#) f2)) is_differentiable_in x by A2, A11, A12, Th86; ::_thesis: verum
end;
then A13: arccot * (f1 + (h (#) f2)) is_differentiable_on Z by A1, A2, FDIFF_1:9;
 for x being Real st x in Z holds
((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) )
assume A14: x in Z ; ::_thesis: ((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)))
then A15: (f1 + (h (#) f2)) . x = (f1 . x) + ((h (#) f2) . x) by A6, VALUED_1:def_1
.= (f1 . x) + (h * (f2 . x)) by A7, A14, VALUED_1:def_5
.= (r + (s * x)) + (h * (f2 . x)) by A4, A14
.= (r + (s * x)) + (h * (x #Z (1 + 1))) by A5, TAYLOR_1:def_1
.= (r + (s * x)) + (h * ((x #Z 1) * (x #Z 1))) by TAYLOR_1:1
.= (r + (s * x)) + (h * (x * (x #Z 1))) by PREPOWER:35
.= (r + (s * x)) + (h * (x ^2)) by PREPOWER:35 ;
A16: f is_differentiable_in x by A2, A8, A14, FDIFF_1:9;
A17: f . x > - 1 by A3, A14;
A18: f . x < 1 by A3, A14;
((arccot * (f1 + (h (#) f2))) `| Z) . x = diff ((arccot * f),x) by A2, A13, A14, FDIFF_1:def_7
.= - ((diff (f,x)) / (1 + ((f . x) ^2))) by A16, A17, A18, Th86
.= - ((((f1 + (h (#) f2)) `| Z) . x) / (1 + ((f . x) ^2))) by A2, A8, A14, FDIFF_1:def_7
.= - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) by A2, A4, A5, A6, A14, A15, FDIFF_4:12 ;
hence  ((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) ; ::_thesis: verum
end;
hence  ( arccot * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * (f1 + (h (#) f2))) `| Z) . x = - ((s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))) ) ) by A1, A2, A9, FDIFF_1:9; ::_thesis: verum
end;

theorem :: SIN_COS9:115
for Z being open Subset of REAL st Z c= dom (arctan * exp_R) & ( for x being Real st x in Z holds
exp_R . x < 1 ) holds
( arctan * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan * exp_R) & ( for x being Real st x in Z holds
exp_R . x < 1 ) implies ( arctan * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) ) ) )
assume that
A1: Z c= dom (arctan * exp_R) and
A2: for x being Real st x in Z holds
exp_R . x < 1 ; ::_thesis: ( arctan * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) ) )
A3: for x being Real st x in Z holds
arctan * exp_R is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arctan * exp_R is_differentiable_in x )
A4: exp_R is_differentiable_in x by SIN_COS:65;
assume  x in Z ; ::_thesis: arctan * exp_R is_differentiable_in x
then A5: exp_R . x < 1 by A2;
 (exp_R . x) + 0 > 0 + (- 1) by SIN_COS:54, XREAL_1:8;
hence  arctan * exp_R is_differentiable_in x by A5, A4, Th85; ::_thesis: verum
end;
then A6: arctan * exp_R is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2))
proof
let x be Real; ::_thesis: ( x in Z implies ((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) )
A7: (exp_R . x) + 0 > 0 + (- 1) by SIN_COS:54, XREAL_1:8;
A8: exp_R is_differentiable_in x by SIN_COS:65;
assume A9: x in Z ; ::_thesis: ((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2))
then A10: exp_R . x < 1 by A2;
((arctan * exp_R) `| Z) . x = diff ((arctan * exp_R),x) by A6, A9, FDIFF_1:def_7
.= (diff (exp_R,x)) / (1 + ((exp_R . x) ^2)) by A7, A10, A8, Th85
.= (exp_R . x) / (1 + ((exp_R . x) ^2)) by SIN_COS:65 ;
hence  ((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) ; ::_thesis: verum
end;
hence  ( arctan * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * exp_R) `| Z) . x = (exp_R . x) / (1 + ((exp_R . x) ^2)) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem :: SIN_COS9:116
for Z being open Subset of REAL st Z c= dom (arccot * exp_R) & ( for x being Real st x in Z holds
exp_R . x < 1 ) holds
( arccot * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arccot * exp_R) & ( for x being Real st x in Z holds
exp_R . x < 1 ) implies ( arccot * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) ) ) )
assume that
A1: Z c= dom (arccot * exp_R) and
A2: for x being Real st x in Z holds
exp_R . x < 1 ; ::_thesis: ( arccot * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) ) )
A3: for x being Real st x in Z holds
arccot * exp_R is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arccot * exp_R is_differentiable_in x )
A4: exp_R is_differentiable_in x by SIN_COS:65;
assume  x in Z ; ::_thesis: arccot * exp_R is_differentiable_in x
then A5: exp_R . x < 1 by A2;
 (exp_R . x) + 0 > 0 + (- 1) by SIN_COS:54, XREAL_1:8;
hence  arccot * exp_R is_differentiable_in x by A5, A4, Th86; ::_thesis: verum
end;
then A6: arccot * exp_R is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) )
A7: (exp_R . x) + 0 > 0 + (- 1) by SIN_COS:54, XREAL_1:8;
A8: exp_R is_differentiable_in x by SIN_COS:65;
assume A9: x in Z ; ::_thesis: ((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2)))
then A10: exp_R . x < 1 by A2;
((arccot * exp_R) `| Z) . x = diff ((arccot * exp_R),x) by A6, A9, FDIFF_1:def_7
.= - ((diff (exp_R,x)) / (1 + ((exp_R . x) ^2))) by A7, A10, A8, Th86
.= - ((exp_R . x) / (1 + ((exp_R . x) ^2))) by SIN_COS:65 ;
hence  ((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) ; ::_thesis: verum
end;
hence  ( arccot * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * exp_R) `| Z) . x = - ((exp_R . x) / (1 + ((exp_R . x) ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem :: SIN_COS9:117
for Z being open Subset of REAL st Z c= dom (arctan * ln) & ( for x being Real st x in Z holds
( ln . x > - 1 & ln . x < 1 ) ) holds
( arctan * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * ln) `| Z) . x = 1 / (x * (1 + ((ln . x) ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan * ln) & ( for x being Real st x in Z holds
( ln . x > - 1 & ln . x < 1 ) ) implies ( arctan * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * ln) `| Z) . x = 1 / (x * (1 + ((ln . x) ^2))) ) ) )
A1: right_open_halfline 0 =  {  g where g is Real : 0 < g  }  by XXREAL_1:230;
assume that
A2: Z c= dom (arctan * ln) and
A3: for x being Real st x in Z holds
( ln . x > - 1 & ln . x < 1 ) ; ::_thesis: ( arctan * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * ln) `| Z) . x = 1 / (x * (1 + ((ln . x) ^2))) ) )
 dom (arctan * ln) c= dom ln by RELAT_1:25;
then A4: Z c= dom ln by A2, XBOOLE_1:1;
A5: for x being Real st x in Z holds
x > 0
proof
let x be Real; ::_thesis: ( x in Z implies x > 0 )
assume  x in Z ; ::_thesis: x > 0
then  x in right_open_halfline 0 by A4, TAYLOR_1:18;
then  ex g being Real st
( x = g & 0 < g ) by A1;
hence  x > 0 ; ::_thesis: verum
end;
A6: for x being Real st x in Z holds
arctan * ln is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arctan * ln is_differentiable_in x )
assume A7: x in Z ; ::_thesis: arctan * ln is_differentiable_in x
then A8: ln . x > - 1 by A3;
A9: ln . x < 1 by A3, A7;
 ln is_differentiable_in x by A5, A7, TAYLOR_1:18;
hence  arctan * ln is_differentiable_in x by A8, A9, Th85; ::_thesis: verum
end;
then A10: arctan * ln is_differentiable_on Z by A2, FDIFF_1:9;
 for x being Real st x in Z holds
((arctan * ln) `| Z) . x = 1 / (x * (1 + ((ln . x) ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((arctan * ln) `| Z) . x = 1 / (x * (1 + ((ln . x) ^2))) )
assume A11: x in Z ; ::_thesis: ((arctan * ln) `| Z) . x = 1 / (x * (1 + ((ln . x) ^2)))
then A12: ln is_differentiable_in x by A5, TAYLOR_1:18;
A13: ln . x < 1 by A3, A11;
A14: ln . x > - 1 by A3, A11;
 x > 0 by A5, A11;
then A15: x in right_open_halfline 0 by A1;
((arctan * ln) `| Z) . x = diff ((arctan * ln),x) by A10, A11, FDIFF_1:def_7
.= (diff (ln,x)) / (1 + ((ln . x) ^2)) by A12, A14, A13, Th85
.= (1 / x) / (1 + ((ln . x) ^2)) by A15, TAYLOR_1:18
.= 1 / (x * (1 + ((ln . x) ^2))) by XCMPLX_1:78 ;
hence  ((arctan * ln) `| Z) . x = 1 / (x * (1 + ((ln . x) ^2))) ; ::_thesis: verum
end;
hence  ( arctan * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * ln) `| Z) . x = 1 / (x * (1 + ((ln . x) ^2))) ) ) by A2, A6, FDIFF_1:9; ::_thesis: verum
end;

theorem :: SIN_COS9:118
for Z being open Subset of REAL st Z c= dom (arccot * ln) & ( for x being Real st x in Z holds
( ln . x > - 1 & ln . x < 1 ) ) holds
( arccot * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * ln) `| Z) . x = - (1 / (x * (1 + ((ln . x) ^2)))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arccot * ln) & ( for x being Real st x in Z holds
( ln . x > - 1 & ln . x < 1 ) ) implies ( arccot * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * ln) `| Z) . x = - (1 / (x * (1 + ((ln . x) ^2)))) ) ) )
A1: right_open_halfline 0 =  {  g where g is Real : 0 < g  }  by XXREAL_1:230;
assume that
A2: Z c= dom (arccot * ln) and
A3: for x being Real st x in Z holds
( ln . x > - 1 & ln . x < 1 ) ; ::_thesis: ( arccot * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * ln) `| Z) . x = - (1 / (x * (1 + ((ln . x) ^2)))) ) )
 dom (arccot * ln) c= dom ln by RELAT_1:25;
then A4: Z c= dom ln by A2, XBOOLE_1:1;
A5: for x being Real st x in Z holds
x > 0
proof
let x be Real; ::_thesis: ( x in Z implies x > 0 )
assume  x in Z ; ::_thesis: x > 0
then  x in right_open_halfline 0 by A4, TAYLOR_1:18;
then  ex g being Real st
( x = g & 0 < g ) by A1;
hence  x > 0 ; ::_thesis: verum
end;
A6: for x being Real st x in Z holds
arccot * ln is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arccot * ln is_differentiable_in x )
assume A7: x in Z ; ::_thesis: arccot * ln is_differentiable_in x
then A8: ln . x > - 1 by A3;
A9: ln . x < 1 by A3, A7;
 ln is_differentiable_in x by A5, A7, TAYLOR_1:18;
hence  arccot * ln is_differentiable_in x by A8, A9, Th86; ::_thesis: verum
end;
then A10: arccot * ln is_differentiable_on Z by A2, FDIFF_1:9;
 for x being Real st x in Z holds
((arccot * ln) `| Z) . x = - (1 / (x * (1 + ((ln . x) ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies ((arccot * ln) `| Z) . x = - (1 / (x * (1 + ((ln . x) ^2)))) )
assume A11: x in Z ; ::_thesis: ((arccot * ln) `| Z) . x = - (1 / (x * (1 + ((ln . x) ^2))))
then A12: ln is_differentiable_in x by A5, TAYLOR_1:18;
A13: ln . x < 1 by A3, A11;
A14: ln . x > - 1 by A3, A11;
 x > 0 by A5, A11;
then A15: x in right_open_halfline 0 by A1;
((arccot * ln) `| Z) . x = diff ((arccot * ln),x) by A10, A11, FDIFF_1:def_7
.= - ((diff (ln,x)) / (1 + ((ln . x) ^2))) by A12, A14, A13, Th86
.= - ((1 / x) / (1 + ((ln . x) ^2))) by A15, TAYLOR_1:18
.= - (1 / (x * (1 + ((ln . x) ^2)))) by XCMPLX_1:78 ;
hence  ((arccot * ln) `| Z) . x = - (1 / (x * (1 + ((ln . x) ^2)))) ; ::_thesis: verum
end;
hence  ( arccot * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * ln) `| Z) . x = - (1 / (x * (1 + ((ln . x) ^2)))) ) ) by A2, A6, FDIFF_1:9; ::_thesis: verum
end;

theorem :: SIN_COS9:119
for Z being open Subset of REAL st Z c= dom (exp_R * arctan) & Z c= ].(- 1),1.[ holds
( exp_R * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * arctan) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (exp_R * arctan) & Z c= ].(- 1),1.[ implies ( exp_R * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * arctan) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2)) ) ) )
assume that
A1: Z c= dom (exp_R * arctan) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( exp_R * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * arctan) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2)) ) )
A3: for x being Real st x in Z holds
exp_R * arctan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies exp_R * arctan is_differentiable_in x )
assume A4: x in Z ; ::_thesis: exp_R * arctan is_differentiable_in x
 arctan is_differentiable_on Z by A2, Th81;
then A5: arctan is_differentiable_in x by A4, FDIFF_1:9;
 exp_R is_differentiable_in arctan . x by SIN_COS:65;
hence  exp_R * arctan is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum
end;
then A6: exp_R * arctan is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((exp_R * arctan) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2))
proof
let x be Real; ::_thesis: ( x in Z implies ((exp_R * arctan) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2)) )
assume A7: x in Z ; ::_thesis: ((exp_R * arctan) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2))
A8: exp_R is_differentiable_in arctan . x by SIN_COS:65;
A9: arctan is_differentiable_on Z by A2, Th81;
then  arctan is_differentiable_in x by A7, FDIFF_1:9;
then  diff ((exp_R * arctan),x) = (diff (exp_R,(arctan . x))) * (diff (arctan,x)) by A8, FDIFF_2:13
.= (diff (exp_R,(arctan . x))) * ((arctan `| Z) . x) by A7, A9, FDIFF_1:def_7
.= (diff (exp_R,(arctan . x))) * (1 / (1 + (x ^2))) by A2, A7, Th81
.= (exp_R . (arctan . x)) / (1 + (x ^2)) by SIN_COS:65 ;
hence  ((exp_R * arctan) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2)) by A6, A7, FDIFF_1:def_7; ::_thesis: verum
end;
hence  ( exp_R * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * arctan) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2)) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem :: SIN_COS9:120
for Z being open Subset of REAL st Z c= dom (exp_R * arccot) & Z c= ].(- 1),1.[ holds
( exp_R * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * arccot) `| Z) . x = - ((exp_R . (arccot . x)) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (exp_R * arccot) & Z c= ].(- 1),1.[ implies ( exp_R * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * arccot) `| Z) . x = - ((exp_R . (arccot . x)) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (exp_R * arccot) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( exp_R * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * arccot) `| Z) . x = - ((exp_R . (arccot . x)) / (1 + (x ^2))) ) )
A3: for x being Real st x in Z holds
exp_R * arccot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies exp_R * arccot is_differentiable_in x )
assume A4: x in Z ; ::_thesis: exp_R * arccot is_differentiable_in x
 arccot is_differentiable_on Z by A2, Th82;
then A5: arccot is_differentiable_in x by A4, FDIFF_1:9;
 exp_R is_differentiable_in arccot . x by SIN_COS:65;
hence  exp_R * arccot is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum
end;
then A6: exp_R * arccot is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((exp_R * arccot) `| Z) . x = - ((exp_R . (arccot . x)) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((exp_R * arccot) `| Z) . x = - ((exp_R . (arccot . x)) / (1 + (x ^2))) )
assume A7: x in Z ; ::_thesis: ((exp_R * arccot) `| Z) . x = - ((exp_R . (arccot . x)) / (1 + (x ^2)))
A8: exp_R is_differentiable_in arccot . x by SIN_COS:65;
A9: arccot is_differentiable_on Z by A2, Th82;
then  arccot is_differentiable_in x by A7, FDIFF_1:9;
then  diff ((exp_R * arccot),x) = (diff (exp_R,(arccot . x))) * (diff (arccot,x)) by A8, FDIFF_2:13
.= (diff (exp_R,(arccot . x))) * ((arccot `| Z) . x) by A7, A9, FDIFF_1:def_7
.= (diff (exp_R,(arccot . x))) * (- (1 / (1 + (x ^2)))) by A2, A7, Th82
.= - ((diff (exp_R,(arccot . x))) * (1 / (1 + (x ^2))))
.= - ((exp_R . (arccot . x)) / (1 + (x ^2))) by SIN_COS:65 ;
hence  ((exp_R * arccot) `| Z) . x = - ((exp_R . (arccot . x)) / (1 + (x ^2))) by A6, A7, FDIFF_1:def_7; ::_thesis: verum
end;
hence  ( exp_R * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * arccot) `| Z) . x = - ((exp_R . (arccot . x)) / (1 + (x ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem :: SIN_COS9:121
for Z being open Subset of REAL st Z c= dom (arctan - (id Z)) & Z c= ].(- 1),1.[ holds
( arctan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan - (id Z)) & Z c= ].(- 1),1.[ implies ( arctan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (arctan - (id Z)) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( arctan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) )
A3: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
 Z c= (dom arctan) /\ (dom (id Z)) by A1, VALUED_1:12;
then A4: Z c= dom (id Z) by XBOOLE_1:18;
then A5: id Z is_differentiable_on Z by A3, FDIFF_1:23;
A6: arctan is_differentiable_on Z by A2, Th81;
 for x being Real st x in Z holds
((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) )
A7: 1 + (x ^2) > 0 by XREAL_1:34, XREAL_1:63;
assume A8: x in Z ; ::_thesis: ((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2)))
then ((arctan - (id Z)) `| Z) . x = (diff (arctan,x)) - (diff ((id Z),x)) by A1, A5, A6, FDIFF_1:19
.= ((arctan `| Z) . x) - (diff ((id Z),x)) by A6, A8, FDIFF_1:def_7
.= (1 / (1 + (x ^2))) - (diff ((id Z),x)) by A2, A8, Th81
.= (1 / (1 + (x ^2))) - (((id Z) `| Z) . x) by A5, A8, FDIFF_1:def_7
.= (1 / (1 + (x ^2))) - 1 by A4, A3, A8, FDIFF_1:23
.= (1 / (1 + (x ^2))) - ((1 + (x ^2)) / (1 + (x ^2))) by A7, XCMPLX_1:60
.= - ((x ^2) / (1 + (x ^2))) ;
hence  ((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( arctan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) ) by A1, A5, A6, FDIFF_1:19; ::_thesis: verum
end;

theorem :: SIN_COS9:122
for Z being open Subset of REAL st Z c= dom ((- arccot) - (id Z)) & Z c= ].(- 1),1.[ holds
( (- arccot) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((- arccot) - (id Z)) & Z c= ].(- 1),1.[ implies ( (- arccot) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom ((- arccot) - (id Z)) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( (- arccot) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) )
A3: arccot is_differentiable_on Z by A2, Th82;
A4: Z c= (dom (- arccot)) /\ (dom (id Z)) by A1, VALUED_1:12;
then A5: Z c= dom (id Z) by XBOOLE_1:18;
A6: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
then A7: id Z is_differentiable_on Z by A5, FDIFF_1:23;
A8: Z c= dom ((- 1) (#) arccot) by A4, XBOOLE_1:18;
then A9: - arccot is_differentiable_on Z by A3, FDIFF_1:20;
 for x being Real st x in Z holds
(((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies (((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) )
A10: 1 + (x ^2) > 0 by XREAL_1:34, XREAL_1:63;
assume A11: x in Z ; ::_thesis: (((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2)))
then (((- arccot) - (id Z)) `| Z) . x = (diff ((- arccot),x)) - (diff ((id Z),x)) by A1, A7, A9, FDIFF_1:19
.= (((- arccot) `| Z) . x) - (diff ((id Z),x)) by A9, A11, FDIFF_1:def_7
.= ((- 1) * (diff (arccot,x))) - (diff ((id Z),x)) by A8, A3, A11, FDIFF_1:20
.= ((- 1) * ((arccot `| Z) . x)) - (diff ((id Z),x)) by A3, A11, FDIFF_1:def_7
.= ((- 1) * (- (1 / (1 + (x ^2))))) - (diff ((id Z),x)) by A2, A11, Th82
.= (1 / (1 + (x ^2))) - (((id Z) `| Z) . x) by A7, A11, FDIFF_1:def_7
.= (1 / (1 + (x ^2))) - 1 by A5, A6, A11, FDIFF_1:23
.= (1 / (1 + (x ^2))) - ((1 + (x ^2)) / (1 + (x ^2))) by A10, XCMPLX_1:60
.= - ((x ^2) / (1 + (x ^2))) ;
hence  (((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( (- arccot) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) ) by A1, A7, A9, FDIFF_1:19; ::_thesis: verum
end;

theorem :: SIN_COS9:123
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( exp_R (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( exp_R (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) ) ) )
assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( exp_R (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) ) )
then A2: arctan is_differentiable_on Z by Th81;
 ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arctan by Th23, XBOOLE_1:1;
then  Z c= dom arctan by A1, XBOOLE_1:1;
then  Z c= (dom exp_R) /\ (dom arctan) by SIN_COS:47, XBOOLE_1:19;
then A3: Z c= dom (exp_R (#) arctan) by VALUED_1:def_4;
 for x being Real st x in Z holds
exp_R is_differentiable_in x by SIN_COS:65;
then A4: exp_R is_differentiable_on Z by FDIFF_1:9, SIN_COS:47;
 for x being Real st x in Z holds
((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) )
assume A5: x in Z ; ::_thesis: ((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2)))
then ((exp_R (#) arctan) `| Z) . x = ((arctan . x) * (diff (exp_R,x))) + ((exp_R . x) * (diff (arctan,x))) by A3, A4, A2, FDIFF_1:21
.= ((arctan . x) * (exp_R . x)) + ((exp_R . x) * (diff (arctan,x))) by SIN_COS:65
.= ((exp_R . x) * (arctan . x)) + ((exp_R . x) * ((arctan `| Z) . x)) by A2, A5, FDIFF_1:def_7
.= ((exp_R . x) * (arctan . x)) + ((exp_R . x) * (1 / (1 + (x ^2)))) by A1, A5, Th81
.= ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) ;
hence  ((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( exp_R (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) ) ) by A3, A4, A2, FDIFF_1:21; ::_thesis: verum
end;

theorem :: SIN_COS9:124
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( exp_R (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( exp_R (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) ) ) )
assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( exp_R (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) ) )
then A2: arccot is_differentiable_on Z by Th82;
 ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by Th24, XBOOLE_1:1;
then  Z c= dom arccot by A1, XBOOLE_1:1;
then  Z c= (dom exp_R) /\ (dom arccot) by SIN_COS:47, XBOOLE_1:19;
then A3: Z c= dom (exp_R (#) arccot) by VALUED_1:def_4;
 for x being Real st x in Z holds
exp_R is_differentiable_in x by SIN_COS:65;
then A4: exp_R is_differentiable_on Z by FDIFF_1:9, SIN_COS:47;
 for x being Real st x in Z holds
((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) )
assume A5: x in Z ; ::_thesis: ((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2)))
then ((exp_R (#) arccot) `| Z) . x = ((arccot . x) * (diff (exp_R,x))) + ((exp_R . x) * (diff (arccot,x))) by A3, A4, A2, FDIFF_1:21
.= ((arccot . x) * (exp_R . x)) + ((exp_R . x) * (diff (arccot,x))) by SIN_COS:65
.= ((exp_R . x) * (arccot . x)) + ((exp_R . x) * ((arccot `| Z) . x)) by A2, A5, FDIFF_1:def_7
.= ((exp_R . x) * (arccot . x)) + ((exp_R . x) * (- (1 / (1 + (x ^2))))) by A1, A5, Th82
.= ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) ;
hence  ((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( exp_R (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) arccot) `| Z) . x = ((exp_R . x) * (arccot . x)) - ((exp_R . x) / (1 + (x ^2))) ) ) by A3, A4, A2, FDIFF_1:21; ::_thesis: verum
end;

theorem :: SIN_COS9:125
for r being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (((1 / r) (#) (arctan * f)) - (id Z)) & ( for x being Real st x in Z holds
( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ) holds
( ((1 / r) (#) (arctan * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) )
proof
let r be Real; ::_thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (((1 / r) (#) (arctan * f)) - (id Z)) & ( for x being Real st x in Z holds
( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ) holds
( ((1 / r) (#) (arctan * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) )
let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (((1 / r) (#) (arctan * f)) - (id Z)) & ( for x being Real st x in Z holds
( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ) holds
( ((1 / r) (#) (arctan * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) )
let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (((1 / r) (#) (arctan * f)) - (id Z)) & ( for x being Real st x in Z holds
( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ) implies ( ((1 / r) (#) (arctan * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) ) )
assume that
A1: Z c= dom (((1 / r) (#) (arctan * f)) - (id Z)) and
A2: for x being Real st x in Z holds
( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ; ::_thesis: ( ((1 / r) (#) (arctan * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) )
A3: for x being Real st x in Z holds
( f . x = (r * x) + 0 & f . x > - 1 & f . x < 1 ) by A2;
set g = (1 / r) (#) (arctan * f);
A4: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
A5: Z c= (dom ((1 / r) (#) (arctan * f))) /\ (dom (id Z)) by A1, VALUED_1:12;
then A6: Z c= dom ((1 / r) (#) (arctan * f)) by XBOOLE_1:18;
A7: Z c= dom (id Z) by A5, XBOOLE_1:18;
then A8: id Z is_differentiable_on Z by A4, FDIFF_1:23;
A9: Z c= dom (arctan * f) by A6, VALUED_1:def_5;
then A10: arctan * f is_differentiable_on Z by A3, Th87;
then A11: (1 / r) (#) (arctan * f) is_differentiable_on Z by A6, FDIFF_1:20;
 for x being Real st x in Z holds
((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) )
A12: 1 + ((r * x) ^2) > 0 by XREAL_1:34, XREAL_1:63;
assume A13: x in Z ; ::_thesis: ((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2)))
then A14: r <> 0 by A2;
((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = (diff (((1 / r) (#) (arctan * f)),x)) - (diff ((id Z),x)) by A1, A11, A8, A13, FDIFF_1:19
.= ((((1 / r) (#) (arctan * f)) `| Z) . x) - (diff ((id Z),x)) by A11, A13, FDIFF_1:def_7
.= ((1 / r) * (diff ((arctan * f),x))) - (diff ((id Z),x)) by A6, A10, A13, FDIFF_1:20
.= ((1 / r) * (((arctan * f) `| Z) . x)) - (diff ((id Z),x)) by A10, A13, FDIFF_1:def_7
.= ((1 / r) * (((arctan * f) `| Z) . x)) - (((id Z) `| Z) . x) by A8, A13, FDIFF_1:def_7
.= ((1 / r) * (r / (1 + (((r * x) + 0) ^2)))) - (((id Z) `| Z) . x) by A3, A9, A13, Th87
.= ((1 / r) * (r / (1 + ((r * x) ^2)))) - 1 by A7, A4, A13, FDIFF_1:23
.= ((1 * r) / (r * (1 + ((r * x) ^2)))) - 1 by XCMPLX_1:76
.= (1 / (1 + ((r * x) ^2))) - 1 by A14, XCMPLX_1:91
.= (1 / (1 + ((r * x) ^2))) - ((1 + ((r * x) ^2)) / (1 + ((r * x) ^2))) by A12, XCMPLX_1:60
.= - (((r * x) ^2) / (1 + ((r * x) ^2))) ;
hence  ((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ; ::_thesis: verum
end;
hence  ( ((1 / r) (#) (arctan * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((1 / r) (#) (arctan * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) ) by A1, A11, A8, FDIFF_1:19; ::_thesis: verum
end;

theorem :: SIN_COS9:126
for r being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (((- (1 / r)) (#) (arccot * f)) - (id Z)) & ( for x being Real st x in Z holds
( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ) holds
( ((- (1 / r)) (#) (arccot * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) )
proof
let r be Real; ::_thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (((- (1 / r)) (#) (arccot * f)) - (id Z)) & ( for x being Real st x in Z holds
( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ) holds
( ((- (1 / r)) (#) (arccot * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) )
let Z be open Subset of REAL; ::_thesis: for f being PartFunc of REAL,REAL st Z c= dom (((- (1 / r)) (#) (arccot * f)) - (id Z)) & ( for x being Real st x in Z holds
( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ) holds
( ((- (1 / r)) (#) (arccot * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) )
let f be PartFunc of REAL,REAL; ::_thesis: ( Z c= dom (((- (1 / r)) (#) (arccot * f)) - (id Z)) & ( for x being Real st x in Z holds
( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ) implies ( ((- (1 / r)) (#) (arccot * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) ) )
assume that
A1: Z c= dom (((- (1 / r)) (#) (arccot * f)) - (id Z)) and
A2: for x being Real st x in Z holds
( f . x = r * x & r <> 0 & f . x > - 1 & f . x < 1 ) ; ::_thesis: ( ((- (1 / r)) (#) (arccot * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) )
A3: for x being Real st x in Z holds
( f . x = (r * x) + 0 & f . x > - 1 & f . x < 1 ) by A2;
set g = (- (1 / r)) (#) (arccot * f);
A4: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
A5: Z c= (dom ((- (1 / r)) (#) (arccot * f))) /\ (dom (id Z)) by A1, VALUED_1:12;
then A6: Z c= dom ((- (1 / r)) (#) (arccot * f)) by XBOOLE_1:18;
A7: Z c= dom (id Z) by A5, XBOOLE_1:18;
then A8: id Z is_differentiable_on Z by A4, FDIFF_1:23;
A9: Z c= dom (arccot * f) by A6, VALUED_1:def_5;
then A10: arccot * f is_differentiable_on Z by A3, Th88;
then A11: (- (1 / r)) (#) (arccot * f) is_differentiable_on Z by A6, FDIFF_1:20;
 for x being Real st x in Z holds
((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) )
A12: 1 + ((r * x) ^2) > 0 by XREAL_1:34, XREAL_1:63;
assume A13: x in Z ; ::_thesis: ((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2)))
then A14: r <> 0 by A2;
((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = (diff (((- (1 / r)) (#) (arccot * f)),x)) - (diff ((id Z),x)) by A1, A11, A8, A13, FDIFF_1:19
.= ((((- (1 / r)) (#) (arccot * f)) `| Z) . x) - (diff ((id Z),x)) by A11, A13, FDIFF_1:def_7
.= ((- (1 / r)) * (diff ((arccot * f),x))) - (diff ((id Z),x)) by A6, A10, A13, FDIFF_1:20
.= ((- (1 / r)) * (((arccot * f) `| Z) . x)) - (diff ((id Z),x)) by A10, A13, FDIFF_1:def_7
.= ((- (1 / r)) * (((arccot * f) `| Z) . x)) - (((id Z) `| Z) . x) by A8, A13, FDIFF_1:def_7
.= ((- (1 / r)) * (- (r / (1 + (((r * x) + 0) ^2))))) - (((id Z) `| Z) . x) by A3, A9, A13, Th88
.= (((- 1) / r) * ((- r) / (1 + ((r * x) ^2)))) - 1 by A7, A4, A13, FDIFF_1:23
.= (((- 1) * (- r)) / (r * (1 + ((r * x) ^2)))) - 1 by XCMPLX_1:76
.= ((1 * r) / (r * (1 + ((r * x) ^2)))) - 1
.= (1 / (1 + ((r * x) ^2))) - 1 by A14, XCMPLX_1:91
.= (1 / (1 + ((r * x) ^2))) - ((1 + ((r * x) ^2)) / (1 + ((r * x) ^2))) by A12, XCMPLX_1:60
.= - (((r * x) ^2) / (1 + ((r * x) ^2))) ;
hence  ((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ; ::_thesis: verum
end;
hence  ( ((- (1 / r)) (#) (arccot * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((- (1 / r)) (#) (arccot * f)) - (id Z)) `| Z) . x = - (((r * x) ^2) / (1 + ((r * x) ^2))) ) ) by A1, A11, A8, FDIFF_1:19; ::_thesis: verum
end;

theorem :: SIN_COS9:127
for Z being open Subset of REAL st Z c= dom (ln (#) arctan) & Z c= ].(- 1),1.[ holds
( ln (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) arctan) `| Z) . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (ln (#) arctan) & Z c= ].(- 1),1.[ implies ( ln (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) arctan) `| Z) . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) ) ) )
A1: right_open_halfline 0 =  {  g where g is Real : 0 < g  }  by XXREAL_1:230;
assume that
A2: Z c= dom (ln (#) arctan) and
A3: Z c= ].(- 1),1.[ ; ::_thesis: ( ln (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) arctan) `| Z) . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) ) )
A4: arctan is_differentiable_on Z by A3, Th81;
 Z c= (dom ln) /\ (dom arctan) by A2, VALUED_1:def_4;
then A5: Z c= dom ln by XBOOLE_1:18;
A6: for x being Real st x in Z holds
x > 0
proof
let x be Real; ::_thesis: ( x in Z implies x > 0 )
assume  x in Z ; ::_thesis: x > 0
then  x in right_open_halfline 0 by A5, TAYLOR_1:18;
then  ex g being Real st
( x = g & 0 < g ) by A1;
hence  x > 0 ; ::_thesis: verum
end;
then  for x being Real st x in Z holds
ln is_differentiable_in x by TAYLOR_1:18;
then A7: ln is_differentiable_on Z by A5, FDIFF_1:9;
A8: for x being Real st x in Z holds
diff (ln,x) = 1 / x
proof
let x be Real; ::_thesis: ( x in Z implies diff (ln,x) = 1 / x )
assume  x in Z ; ::_thesis: diff (ln,x) = 1 / x
then  x > 0 by A6;
then  x in right_open_halfline 0 by A1;
hence  diff (ln,x) = 1 / x by TAYLOR_1:18; ::_thesis: verum
end;
 for x being Real st x in Z holds
((ln (#) arctan) `| Z) . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((ln (#) arctan) `| Z) . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) )
assume A9: x in Z ; ::_thesis: ((ln (#) arctan) `| Z) . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2)))
then ((ln (#) arctan) `| Z) . x = ((arctan . x) * (diff (ln,x))) + ((ln . x) * (diff (arctan,x))) by A2, A7, A4, FDIFF_1:21
.= ((arctan . x) * (1 / x)) + ((ln . x) * (diff (arctan,x))) by A8, A9
.= ((arctan . x) * (1 / x)) + ((ln . x) * ((arctan `| Z) . x)) by A4, A9, FDIFF_1:def_7
.= (((arctan . x) * 1) / x) + ((ln . x) * (1 / (1 + (x ^2)))) by A3, A9, Th81
.= ((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) ;
hence  ((ln (#) arctan) `| Z) . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( ln (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) arctan) `| Z) . x = ((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) ) ) by A2, A7, A4, FDIFF_1:21; ::_thesis: verum
end;

theorem :: SIN_COS9:128
for Z being open Subset of REAL st Z c= dom (ln (#) arccot) & Z c= ].(- 1),1.[ holds
( ln (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (ln (#) arccot) & Z c= ].(- 1),1.[ implies ( ln (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) ) ) )
A1: right_open_halfline 0 =  {  g where g is Real : 0 < g  }  by XXREAL_1:230;
assume that
A2: Z c= dom (ln (#) arccot) and
A3: Z c= ].(- 1),1.[ ; ::_thesis: ( ln (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) ) )
A4: arccot is_differentiable_on Z by A3, Th82;
 Z c= (dom ln) /\ (dom arccot) by A2, VALUED_1:def_4;
then A5: Z c= dom ln by XBOOLE_1:18;
A6: for x being Real st x in Z holds
x > 0
proof
let x be Real; ::_thesis: ( x in Z implies x > 0 )
assume  x in Z ; ::_thesis: x > 0
then  x in right_open_halfline 0 by A5, TAYLOR_1:18;
then  ex g being Real st
( x = g & 0 < g ) by A1;
hence  x > 0 ; ::_thesis: verum
end;
then  for x being Real st x in Z holds
ln is_differentiable_in x by TAYLOR_1:18;
then A7: ln is_differentiable_on Z by A5, FDIFF_1:9;
A8: for x being Real st x in Z holds
diff (ln,x) = 1 / x
proof
let x be Real; ::_thesis: ( x in Z implies diff (ln,x) = 1 / x )
assume  x in Z ; ::_thesis: diff (ln,x) = 1 / x
then  x > 0 by A6;
then  x in right_open_halfline 0 by A1;
hence  diff (ln,x) = 1 / x by TAYLOR_1:18; ::_thesis: verum
end;
 for x being Real st x in Z holds
((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) )
assume A9: x in Z ; ::_thesis: ((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2)))
then ((ln (#) arccot) `| Z) . x = ((arccot . x) * (diff (ln,x))) + ((ln . x) * (diff (arccot,x))) by A2, A7, A4, FDIFF_1:21
.= ((arccot . x) * (1 / x)) + ((ln . x) * (diff (arccot,x))) by A8, A9
.= ((arccot . x) * (1 / x)) + ((ln . x) * ((arccot `| Z) . x)) by A4, A9, FDIFF_1:def_7
.= ((arccot . x) * (1 / x)) + ((ln . x) * (- (1 / (1 + (x ^2))))) by A3, A9, Th82
.= ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) ;
hence  ((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( ln (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) arccot) `| Z) . x = ((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) ) ) by A2, A7, A4, FDIFF_1:21; ::_thesis: verum
end;

theorem :: SIN_COS9:129
for Z being open Subset of REAL st not 0 in Z & Z c= dom (((id Z) ^) (#) arctan) & Z c= ].(- 1),1.[ holds
( ((id Z) ^) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( not 0 in Z & Z c= dom (((id Z) ^) (#) arctan) & Z c= ].(- 1),1.[ implies ( ((id Z) ^) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ) ) )
set f = id Z;
assume that
A1: not 0 in Z and
A2: Z c= dom (((id Z) ^) (#) arctan) and
A3: Z c= ].(- 1),1.[ ; ::_thesis: ( ((id Z) ^) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ) )
A4: (id Z) ^ is_differentiable_on Z by A1, FDIFF_5:4;
A5: arctan is_differentiable_on Z by A3, Th81;
 Z c= (dom ((id Z) ^)) /\ (dom arctan) by A2, VALUED_1:def_4;
then A6: Z c= dom ((id Z) ^) by XBOOLE_1:18;
 for x being Real st x in Z holds
((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies ((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) )
assume A7: x in Z ; ::_thesis: ((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2))))
then ((((id Z) ^) (#) arctan) `| Z) . x = ((arctan . x) * (diff (((id Z) ^),x))) + ((((id Z) ^) . x) * (diff (arctan,x))) by A2, A4, A5, FDIFF_1:21
.= ((arctan . x) * ((((id Z) ^) `| Z) . x)) + ((((id Z) ^) . x) * (diff (arctan,x))) by A4, A7, FDIFF_1:def_7
.= ((arctan . x) * (- (1 / (x ^2)))) + ((((id Z) ^) . x) * (diff (arctan,x))) by A1, A7, FDIFF_5:4
.= (- ((arctan . x) * (1 / (x ^2)))) + ((((id Z) ^) . x) * ((arctan `| Z) . x)) by A5, A7, FDIFF_1:def_7
.= (- (((arctan . x) * 1) / (x ^2))) + ((((id Z) ^) . x) * (1 / (1 + (x ^2)))) by A3, A7, Th81
.= (- ((arctan . x) / (x ^2))) + ((((id Z) . x) ") * (1 / (1 + (x ^2)))) by A6, A7, RFUNCT_1:def_2
.= (- ((arctan . x) / (x ^2))) + ((1 / x) * (1 / (1 + (x ^2)))) by A7, FUNCT_1:18
.= (- ((arctan . x) / (x ^2))) + ((1 * 1) / (x * (1 + (x ^2)))) by XCMPLX_1:76
.= (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ;
hence  ((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ; ::_thesis: verum
end;
hence  ( ((id Z) ^) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ) ) by A2, A4, A5, FDIFF_1:21; ::_thesis: verum
end;

theorem :: SIN_COS9:130
for Z being open Subset of REAL st not 0 in Z & Z c= dom (((id Z) ^) (#) arccot) & Z c= ].(- 1),1.[ holds
( ((id Z) ^) (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) arccot) `| Z) . x = (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( not 0 in Z & Z c= dom (((id Z) ^) (#) arccot) & Z c= ].(- 1),1.[ implies ( ((id Z) ^) (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) arccot) `| Z) . x = (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) ) ) )
set f = id Z;
assume that
A1: not 0 in Z and
A2: Z c= dom (((id Z) ^) (#) arccot) and
A3: Z c= ].(- 1),1.[ ; ::_thesis: ( ((id Z) ^) (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) arccot) `| Z) . x = (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) ) )
A4: (id Z) ^ is_differentiable_on Z by A1, FDIFF_5:4;
A5: arccot is_differentiable_on Z by A3, Th82;
 Z c= (dom ((id Z) ^)) /\ (dom arccot) by A2, VALUED_1:def_4;
then A6: Z c= dom ((id Z) ^) by XBOOLE_1:18;
 for x being Real st x in Z holds
((((id Z) ^) (#) arccot) `| Z) . x = (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies ((((id Z) ^) (#) arccot) `| Z) . x = (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) )
assume A7: x in Z ; ::_thesis: ((((id Z) ^) (#) arccot) `| Z) . x = (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2))))
then ((((id Z) ^) (#) arccot) `| Z) . x = ((arccot . x) * (diff (((id Z) ^),x))) + ((((id Z) ^) . x) * (diff (arccot,x))) by A2, A4, A5, FDIFF_1:21
.= ((arccot . x) * ((((id Z) ^) `| Z) . x)) + ((((id Z) ^) . x) * (diff (arccot,x))) by A4, A7, FDIFF_1:def_7
.= ((arccot . x) * (- (1 / (x ^2)))) + ((((id Z) ^) . x) * (diff (arccot,x))) by A1, A7, FDIFF_5:4
.= (- ((arccot . x) * (1 / (x ^2)))) + ((((id Z) ^) . x) * ((arccot `| Z) . x)) by A5, A7, FDIFF_1:def_7
.= (- ((arccot . x) * (1 / (x ^2)))) + ((((id Z) ^) . x) * (- (1 / (1 + (x ^2))))) by A3, A7, Th82
.= (- (((arccot . x) * 1) / (x ^2))) - ((((id Z) ^) . x) * (1 / (1 + (x ^2))))
.= (- ((arccot . x) / (x ^2))) - ((((id Z) . x) ") * (1 / (1 + (x ^2)))) by A6, A7, RFUNCT_1:def_2
.= (- ((arccot . x) / (x ^2))) - ((1 / x) * (1 / (1 + (x ^2)))) by A7, FUNCT_1:18
.= (- ((arccot . x) / (x ^2))) - ((1 * 1) / (x * (1 + (x ^2)))) by XCMPLX_1:76
.= (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) ;
hence  ((((id Z) ^) (#) arccot) `| Z) . x = (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) ; ::_thesis: verum
end;
hence  ( ((id Z) ^) (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) arccot) `| Z) . x = (- ((arccot . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) ) ) by A2, A4, A5, FDIFF_1:21; ::_thesis: verum
end;