:: FDIFF_11 semantic presentation

begin


theorem :: FDIFF_11:1
for Z being open Subset of REAL st Z c= dom (arctan * sin) & ( for x being Real st x in Z holds
( sin . x > - 1 & sin . x < 1 ) ) holds
( arctan * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan * sin) & ( for x being Real st x in Z holds
( sin . x > - 1 & sin . x < 1 ) ) implies ( arctan * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) ) ) )
assume that
A1: Z c= dom (arctan * sin) and
A2: for x being Real st x in Z holds
( sin . x > - 1 & sin . x < 1 ) ; ::_thesis: ( arctan * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) ) )
A3: for x being Real st x in Z holds
arctan * sin is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arctan * sin is_differentiable_in x )
assume  x in Z ; ::_thesis: arctan * sin is_differentiable_in x
then A4: ( sin . x > - 1 & sin . x < 1 ) by A2;
 sin is_differentiable_in x by SIN_COS:64;
hence  arctan * sin is_differentiable_in x by A4, SIN_COS9:85; ::_thesis: verum
end;
then A5: arctan * sin is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2))
proof
let x be Real; ::_thesis: ( x in Z implies ((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) )
A6: sin is_differentiable_in x by SIN_COS:64;
assume A7: x in Z ; ::_thesis: ((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2))
then A8: ( sin . x > - 1 & sin . x < 1 ) by A2;
((arctan * sin) `| Z) . x = diff ((arctan * sin),x) by A5, A7, FDIFF_1:def_7
.= (diff (sin,x)) / (1 + ((sin . x) ^2)) by A6, A8, SIN_COS9:85
.= (cos . x) / (1 + ((sin . x) ^2)) by SIN_COS:64 ;
hence  ((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) ; ::_thesis: verum
end;
hence  ( arctan * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:2
for Z being open Subset of REAL st Z c= dom (arccot * sin) & ( for x being Real st x in Z holds
( sin . x > - 1 & sin . x < 1 ) ) holds
( arccot * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * sin) `| Z) . x = - ((cos . x) / (1 + ((sin . x) ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arccot * sin) & ( for x being Real st x in Z holds
( sin . x > - 1 & sin . x < 1 ) ) implies ( arccot * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * sin) `| Z) . x = - ((cos . x) / (1 + ((sin . x) ^2))) ) ) )
assume that
A1: Z c= dom (arccot * sin) and
A2: for x being Real st x in Z holds
( sin . x > - 1 & sin . x < 1 ) ; ::_thesis: ( arccot * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * sin) `| Z) . x = - ((cos . x) / (1 + ((sin . x) ^2))) ) )
A3: for x being Real st x in Z holds
arccot * sin is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arccot * sin is_differentiable_in x )
assume  x in Z ; ::_thesis: arccot * sin is_differentiable_in x
then A4: ( sin . x > - 1 & sin . x < 1 ) by A2;
 sin is_differentiable_in x by SIN_COS:64;
hence  arccot * sin is_differentiable_in x by A4, SIN_COS9:86; ::_thesis: verum
end;
then A5: arccot * sin is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((arccot * sin) `| Z) . x = - ((cos . x) / (1 + ((sin . x) ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((arccot * sin) `| Z) . x = - ((cos . x) / (1 + ((sin . x) ^2))) )
A6: sin is_differentiable_in x by SIN_COS:64;
assume A7: x in Z ; ::_thesis: ((arccot * sin) `| Z) . x = - ((cos . x) / (1 + ((sin . x) ^2)))
then A8: ( sin . x > - 1 & sin . x < 1 ) by A2;
((arccot * sin) `| Z) . x = diff ((arccot * sin),x) by A5, A7, FDIFF_1:def_7
.= - ((diff (sin,x)) / (1 + ((sin . x) ^2))) by A6, A8, SIN_COS9:86
.= - ((cos . x) / (1 + ((sin . x) ^2))) by SIN_COS:64 ;
hence  ((arccot * sin) `| Z) . x = - ((cos . x) / (1 + ((sin . x) ^2))) ; ::_thesis: verum
end;
hence  ( arccot * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * sin) `| Z) . x = - ((cos . x) / (1 + ((sin . x) ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:3
for Z being open Subset of REAL st Z c= dom (arctan * cos) & ( for x being Real st x in Z holds
( cos . x > - 1 & cos . x < 1 ) ) holds
( arctan * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * cos) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan * cos) & ( for x being Real st x in Z holds
( cos . x > - 1 & cos . x < 1 ) ) implies ( arctan * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * cos) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2))) ) ) )
assume that
A1: Z c= dom (arctan * cos) and
A2: for x being Real st x in Z holds
( cos . x > - 1 & cos . x < 1 ) ; ::_thesis: ( arctan * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * cos) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2))) ) )
A3: for x being Real st x in Z holds
arctan * cos is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arctan * cos is_differentiable_in x )
assume  x in Z ; ::_thesis: arctan * cos is_differentiable_in x
then A4: ( cos . x > - 1 & cos . x < 1 ) by A2;
 cos is_differentiable_in x by SIN_COS:63;
hence  arctan * cos is_differentiable_in x by A4, SIN_COS9:85; ::_thesis: verum
end;
then A5: arctan * cos is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((arctan * cos) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((arctan * cos) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2))) )
A6: cos is_differentiable_in x by SIN_COS:63;
assume A7: x in Z ; ::_thesis: ((arctan * cos) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2)))
then A8: ( cos . x > - 1 & cos . x < 1 ) by A2;
((arctan * cos) `| Z) . x = diff ((arctan * cos),x) by A5, A7, FDIFF_1:def_7
.= (diff (cos,x)) / (1 + ((cos . x) ^2)) by A6, A8, SIN_COS9:85
.= (- (sin . x)) / (1 + ((cos . x) ^2)) by SIN_COS:63
.= - ((sin . x) / (1 + ((cos . x) ^2))) ;
hence  ((arctan * cos) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2))) ; ::_thesis: verum
end;
hence  ( arctan * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * cos) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:4
for Z being open Subset of REAL st Z c= dom (arccot * cos) & ( for x being Real st x in Z holds
( cos . x > - 1 & cos . x < 1 ) ) holds
( arccot * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * cos) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arccot * cos) & ( for x being Real st x in Z holds
( cos . x > - 1 & cos . x < 1 ) ) implies ( arccot * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * cos) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2)) ) ) )
assume that
A1: Z c= dom (arccot * cos) and
A2: for x being Real st x in Z holds
( cos . x > - 1 & cos . x < 1 ) ; ::_thesis: ( arccot * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * cos) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2)) ) )
A3: for x being Real st x in Z holds
arccot * cos is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arccot * cos is_differentiable_in x )
assume  x in Z ; ::_thesis: arccot * cos is_differentiable_in x
then A4: ( cos . x > - 1 & cos . x < 1 ) by A2;
 cos is_differentiable_in x by SIN_COS:63;
hence  arccot * cos is_differentiable_in x by A4, SIN_COS9:86; ::_thesis: verum
end;
then A5: arccot * cos is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((arccot * cos) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2))
proof
let x be Real; ::_thesis: ( x in Z implies ((arccot * cos) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2)) )
A6: cos is_differentiable_in x by SIN_COS:63;
assume A7: x in Z ; ::_thesis: ((arccot * cos) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2))
then A8: ( cos . x > - 1 & cos . x < 1 ) by A2;
((arccot * cos) `| Z) . x = diff ((arccot * cos),x) by A5, A7, FDIFF_1:def_7
.= - ((diff (cos,x)) / (1 + ((cos . x) ^2))) by A6, A8, SIN_COS9:86
.= - ((- (sin . x)) / (1 + ((cos . x) ^2))) by SIN_COS:63
.= (sin . x) / (1 + ((cos . x) ^2)) ;
hence  ((arccot * cos) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2)) ; ::_thesis: verum
end;
hence  ( arccot * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * cos) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2)) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:5
for Z being open Subset of REAL st Z c= dom (arctan * tan) & ( for x being Real st x in Z holds
( tan . x > - 1 & tan . x < 1 ) ) holds
( arctan * tan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * tan) `| Z) . x = 1 ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan * tan) & ( for x being Real st x in Z holds
( tan . x > - 1 & tan . x < 1 ) ) implies ( arctan * tan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * tan) `| Z) . x = 1 ) ) )
assume that
A1: Z c= dom (arctan * tan) and
A2: for x being Real st x in Z holds
( tan . x > - 1 & tan . x < 1 ) ; ::_thesis: ( arctan * tan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * tan) `| Z) . x = 1 ) )
 dom (arctan * tan) c= dom tan by RELAT_1:25;
then A3: Z c= dom tan by A1, XBOOLE_1:1;
A4: for x being Real st x in Z holds
arctan * tan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arctan * tan is_differentiable_in x )
assume A5: x in Z ; ::_thesis: arctan * tan is_differentiable_in x
then  cos . x <> 0 by A3, FDIFF_8:1;
then A6: tan is_differentiable_in x by FDIFF_7:46;
 ( tan . x > - 1 & tan . x < 1 ) by A2, A5;
hence  arctan * tan is_differentiable_in x by A6, SIN_COS9:85; ::_thesis: verum
end;
then A7: arctan * tan is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((arctan * tan) `| Z) . x = 1
proof
let x be Real; ::_thesis: ( x in Z implies ((arctan * tan) `| Z) . x = 1 )
assume A8: x in Z ; ::_thesis: ((arctan * tan) `| Z) . x = 1
then A9: ( tan . x > - 1 & tan . x < 1 ) by A2;
A10: tan . x = (sin . x) / (cos . x) by A3, A8, RFUNCT_1:def_1;
A11: cos . x <> 0 by A3, A8, FDIFF_8:1;
then A12: tan is_differentiable_in x by FDIFF_7:46;
A13: (cos . x) ^2 <> 0 by A11, SQUARE_1:12;
((arctan * tan) `| Z) . x = diff ((arctan * tan),x) by A7, A8, FDIFF_1:def_7
.= (diff (tan,x)) / (1 + ((tan . x) ^2)) by A12, A9, SIN_COS9:85
.= (1 / ((cos . x) ^2)) / (1 + ((tan . x) ^2)) by A11, FDIFF_7:46
.= 1 / (((cos . x) ^2) * (1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x))))) by A10, XCMPLX_1:78
.= 1 / (((cos . x) ^2) * (1 + (((sin . x) ^2) / ((cos . x) ^2)))) by XCMPLX_1:76
.= 1 / (((cos . x) ^2) + ((((cos . x) ^2) * ((sin . x) ^2)) / ((cos . x) ^2)))
.= 1 / (((cos . x) ^2) + ((sin . x) ^2)) by A13, XCMPLX_1:89
.= 1 / 1 by SIN_COS:28
.= 1 ;
hence  ((arctan * tan) `| Z) . x = 1 ; ::_thesis: verum
end;
hence  ( arctan * tan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * tan) `| Z) . x = 1 ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:6
for Z being open Subset of REAL st Z c= dom (arccot * tan) & ( for x being Real st x in Z holds
( tan . x > - 1 & tan . x < 1 ) ) holds
( arccot * tan is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * tan) `| Z) . x = - 1 ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arccot * tan) & ( for x being Real st x in Z holds
( tan . x > - 1 & tan . x < 1 ) ) implies ( arccot * tan is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * tan) `| Z) . x = - 1 ) ) )
assume that
A1: Z c= dom (arccot * tan) and
A2: for x being Real st x in Z holds
( tan . x > - 1 & tan . x < 1 ) ; ::_thesis: ( arccot * tan is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * tan) `| Z) . x = - 1 ) )
 dom (arccot * tan) c= dom tan by RELAT_1:25;
then A3: Z c= dom tan by A1, XBOOLE_1:1;
A4: for x being Real st x in Z holds
arccot * tan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arccot * tan is_differentiable_in x )
assume A5: x in Z ; ::_thesis: arccot * tan is_differentiable_in x
then  cos . x <> 0 by A3, FDIFF_8:1;
then A6: tan is_differentiable_in x by FDIFF_7:46;
 ( tan . x > - 1 & tan . x < 1 ) by A2, A5;
hence  arccot * tan is_differentiable_in x by A6, SIN_COS9:86; ::_thesis: verum
end;
then A7: arccot * tan is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((arccot * tan) `| Z) . x = - 1
proof
let x be Real; ::_thesis: ( x in Z implies ((arccot * tan) `| Z) . x = - 1 )
assume A8: x in Z ; ::_thesis: ((arccot * tan) `| Z) . x = - 1
then A9: ( tan . x > - 1 & tan . x < 1 ) by A2;
A10: tan . x = (sin . x) / (cos . x) by A3, A8, RFUNCT_1:def_1;
A11: cos . x <> 0 by A3, A8, FDIFF_8:1;
then A12: tan is_differentiable_in x by FDIFF_7:46;
A13: (cos . x) ^2 <> 0 by A11, SQUARE_1:12;
((arccot * tan) `| Z) . x = diff ((arccot * tan),x) by A7, A8, FDIFF_1:def_7
.= - ((diff (tan,x)) / (1 + ((tan . x) ^2))) by A12, A9, SIN_COS9:86
.= - ((1 / ((cos . x) ^2)) / (1 + ((tan . x) ^2))) by A11, FDIFF_7:46
.= - (1 / (((cos . x) ^2) * (1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x)))))) by A10, XCMPLX_1:78
.= - (1 / (((cos . x) ^2) * (1 + (((sin . x) ^2) / ((cos . x) ^2))))) by XCMPLX_1:76
.= - (1 / (((cos . x) ^2) + ((((cos . x) ^2) * ((sin . x) ^2)) / ((cos . x) ^2))))
.= - (1 / (((cos . x) ^2) + ((sin . x) ^2))) by A13, XCMPLX_1:89
.= - (1 / 1) by SIN_COS:28
.= - 1 ;
hence  ((arccot * tan) `| Z) . x = - 1 ; ::_thesis: verum
end;
hence  ( arccot * tan is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * tan) `| Z) . x = - 1 ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:7
for Z being open Subset of REAL st Z c= dom (arctan * cot) & ( for x being Real st x in Z holds
( cot . x > - 1 & cot . x < 1 ) ) holds
( arctan * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * cot) `| Z) . x = - 1 ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan * cot) & ( for x being Real st x in Z holds
( cot . x > - 1 & cot . x < 1 ) ) implies ( arctan * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * cot) `| Z) . x = - 1 ) ) )
assume that
A1: Z c= dom (arctan * cot) and
A2: for x being Real st x in Z holds
( cot . x > - 1 & cot . x < 1 ) ; ::_thesis: ( arctan * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * cot) `| Z) . x = - 1 ) )
 dom (arctan * cot) c= dom cot by RELAT_1:25;
then A3: Z c= dom cot by A1, XBOOLE_1:1;
A4: for x being Real st x in Z holds
arctan * cot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arctan * cot is_differentiable_in x )
assume A5: x in Z ; ::_thesis: arctan * cot is_differentiable_in x
then  sin . x <> 0 by A3, FDIFF_8:2;
then A6: cot is_differentiable_in x by FDIFF_7:47;
 ( cot . x > - 1 & cot . x < 1 ) by A2, A5;
hence  arctan * cot is_differentiable_in x by A6, SIN_COS9:85; ::_thesis: verum
end;
then A7: arctan * cot is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((arctan * cot) `| Z) . x = - 1
proof
let x be Real; ::_thesis: ( x in Z implies ((arctan * cot) `| Z) . x = - 1 )
assume A8: x in Z ; ::_thesis: ((arctan * cot) `| Z) . x = - 1
then A9: ( cot . x > - 1 & cot . x < 1 ) by A2;
A10: cot . x = (cos . x) / (sin . x) by A3, A8, RFUNCT_1:def_1;
A11: sin . x <> 0 by A3, A8, FDIFF_8:2;
then A12: cot is_differentiable_in x by FDIFF_7:47;
A13: (sin . x) ^2 <> 0 by A11, SQUARE_1:12;
((arctan * cot) `| Z) . x = diff ((arctan * cot),x) by A7, A8, FDIFF_1:def_7
.= (diff (cot,x)) / (1 + ((cot . x) ^2)) by A12, A9, SIN_COS9:85
.= (- (1 / ((sin . x) ^2))) / (1 + ((cot . x) ^2)) by A11, FDIFF_7:47
.= - ((1 / ((sin . x) ^2)) / (1 + ((cot . x) ^2)))
.= - (1 / (((sin . x) ^2) * (1 + (((cos . x) / (sin . x)) * ((cos . x) / (sin . x)))))) by A10, XCMPLX_1:78
.= - (1 / (((sin . x) ^2) * (1 + (((cos . x) ^2) / ((sin . x) ^2))))) by XCMPLX_1:76
.= - (1 / (((sin . x) ^2) + ((((sin . x) ^2) * ((cos . x) ^2)) / ((sin . x) ^2))))
.= - (1 / (((sin . x) ^2) + ((cos . x) ^2))) by A13, XCMPLX_1:89
.= - (1 / 1) by SIN_COS:28
.= - 1 ;
hence  ((arctan * cot) `| Z) . x = - 1 ; ::_thesis: verum
end;
hence  ( arctan * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * cot) `| Z) . x = - 1 ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:8
for Z being open Subset of REAL st Z c= dom (arccot * cot) & ( for x being Real st x in Z holds
( cot . x > - 1 & cot . x < 1 ) ) holds
( arccot * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * cot) `| Z) . x = 1 ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arccot * cot) & ( for x being Real st x in Z holds
( cot . x > - 1 & cot . x < 1 ) ) implies ( arccot * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * cot) `| Z) . x = 1 ) ) )
assume that
A1: Z c= dom (arccot * cot) and
A2: for x being Real st x in Z holds
( cot . x > - 1 & cot . x < 1 ) ; ::_thesis: ( arccot * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * cot) `| Z) . x = 1 ) )
 dom (arccot * cot) c= dom cot by RELAT_1:25;
then A3: Z c= dom cot by A1, XBOOLE_1:1;
A4: for x being Real st x in Z holds
arccot * cot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arccot * cot is_differentiable_in x )
assume A5: x in Z ; ::_thesis: arccot * cot is_differentiable_in x
then  sin . x <> 0 by A3, FDIFF_8:2;
then A6: cot is_differentiable_in x by FDIFF_7:47;
 ( cot . x > - 1 & cot . x < 1 ) by A2, A5;
hence  arccot * cot is_differentiable_in x by A6, SIN_COS9:86; ::_thesis: verum
end;
then A7: arccot * cot is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((arccot * cot) `| Z) . x = 1
proof
let x be Real; ::_thesis: ( x in Z implies ((arccot * cot) `| Z) . x = 1 )
assume A8: x in Z ; ::_thesis: ((arccot * cot) `| Z) . x = 1
then A9: ( cot . x > - 1 & cot . x < 1 ) by A2;
A10: cot . x = (cos . x) / (sin . x) by A3, A8, RFUNCT_1:def_1;
A11: sin . x <> 0 by A3, A8, FDIFF_8:2;
then A12: cot is_differentiable_in x by FDIFF_7:47;
A13: (sin . x) ^2 <> 0 by A11, SQUARE_1:12;
((arccot * cot) `| Z) . x = diff ((arccot * cot),x) by A7, A8, FDIFF_1:def_7
.= - ((diff (cot,x)) / (1 + ((cot . x) ^2))) by A12, A9, SIN_COS9:86
.= - ((- (1 / ((sin . x) ^2))) / (1 + ((cot . x) ^2))) by A11, FDIFF_7:47
.= (1 / ((sin . x) ^2)) / (1 + ((cot . x) ^2))
.= 1 / (((sin . x) ^2) * (1 + (((cos . x) / (sin . x)) * ((cos . x) / (sin . x))))) by A10, XCMPLX_1:78
.= 1 / (((sin . x) ^2) * (1 + (((cos . x) ^2) / ((sin . x) ^2)))) by XCMPLX_1:76
.= 1 / (((sin . x) ^2) + ((((sin . x) ^2) * ((cos . x) ^2)) / ((sin . x) ^2)))
.= 1 / (((sin . x) ^2) + ((cos . x) ^2)) by A13, XCMPLX_1:89
.= 1 / 1 by SIN_COS:28
.= 1 ;
hence  ((arccot * cot) `| Z) . x = 1 ; ::_thesis: verum
end;
hence  ( arccot * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * cot) `| Z) . x = 1 ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:9
for Z being open Subset of REAL st Z c= dom (arctan * arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arctan . x > - 1 & arctan . x < 1 ) ) holds
( arctan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan * arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arctan . x > - 1 & arctan . x < 1 ) ) implies ( arctan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) ) ) )
assume that
A1: Z c= dom (arctan * arctan) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
( arctan . x > - 1 & arctan . x < 1 ) ; ::_thesis: ( arctan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) ) )
A4: for x being Real st x in Z holds
arctan * arctan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arctan * arctan is_differentiable_in x )
assume A5: x in Z ; ::_thesis: arctan * arctan is_differentiable_in x
then A6: ( arctan . x > - 1 & arctan . x < 1 ) by A3;
 arctan is_differentiable_on Z by A2, SIN_COS9:81;
then  arctan is_differentiable_in x by A5, FDIFF_1:9;
hence  arctan * arctan is_differentiable_in x by A6, SIN_COS9:85; ::_thesis: verum
end;
then A7: arctan * arctan is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((arctan * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((arctan * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) )
assume A8: x in Z ; ::_thesis: ((arctan * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))
then A9: ( arctan . x > - 1 & arctan . x < 1 ) by A3;
A10: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A11: arctan is_differentiable_in x by A8, FDIFF_1:9;
((arctan * arctan) `| Z) . x = diff ((arctan * arctan),x) by A7, A8, FDIFF_1:def_7
.= (diff (arctan,x)) / (1 + ((arctan . x) ^2)) by A11, A9, SIN_COS9:85
.= ((arctan `| Z) . x) / (1 + ((arctan . x) ^2)) by A8, A10, FDIFF_1:def_7
.= (1 / (1 + (x ^2))) / (1 + ((arctan . x) ^2)) by A2, A8, SIN_COS9:81
.= 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) by XCMPLX_1:78 ;
hence  ((arctan * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) ; ::_thesis: verum
end;
hence  ( arctan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:10
for Z being open Subset of REAL st Z c= dom (arccot * arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arctan . x > - 1 & arctan . x < 1 ) ) holds
( arccot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * arctan) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arccot * arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arctan . x > - 1 & arctan . x < 1 ) ) implies ( arccot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * arctan) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) ) ) )
assume that
A1: Z c= dom (arccot * arctan) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
( arctan . x > - 1 & arctan . x < 1 ) ; ::_thesis: ( arccot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * arctan) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) ) )
A4: for x being Real st x in Z holds
arccot * arctan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arccot * arctan is_differentiable_in x )
assume A5: x in Z ; ::_thesis: arccot * arctan is_differentiable_in x
then A6: ( arctan . x > - 1 & arctan . x < 1 ) by A3;
 arctan is_differentiable_on Z by A2, SIN_COS9:81;
then  arctan is_differentiable_in x by A5, FDIFF_1:9;
hence  arccot * arctan is_differentiable_in x by A6, SIN_COS9:86; ::_thesis: verum
end;
then A7: arccot * arctan is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((arccot * arctan) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies ((arccot * arctan) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) )
assume A8: x in Z ; ::_thesis: ((arccot * arctan) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2))))
then A9: ( arctan . x > - 1 & arctan . x < 1 ) by A3;
A10: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A11: arctan is_differentiable_in x by A8, FDIFF_1:9;
((arccot * arctan) `| Z) . x = diff ((arccot * arctan),x) by A7, A8, FDIFF_1:def_7
.= - ((diff (arctan,x)) / (1 + ((arctan . x) ^2))) by A11, A9, SIN_COS9:86
.= - (((arctan `| Z) . x) / (1 + ((arctan . x) ^2))) by A8, A10, FDIFF_1:def_7
.= - ((1 / (1 + (x ^2))) / (1 + ((arctan . x) ^2))) by A2, A8, SIN_COS9:81
.= - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) by XCMPLX_1:78 ;
hence  ((arccot * arctan) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) ; ::_thesis: verum
end;
hence  ( arccot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * arctan) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arctan . x) ^2)))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:11
for Z being open Subset of REAL st Z c= dom (arctan * arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arccot . x > - 1 & arccot . x < 1 ) ) holds
( arctan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arctan * arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arccot . x > - 1 & arccot . x < 1 ) ) implies ( arctan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) ) ) )
assume that
A1: Z c= dom (arctan * arccot) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
( arccot . x > - 1 & arccot . x < 1 ) ; ::_thesis: ( arctan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) ) )
A4: for x being Real st x in Z holds
arctan * arccot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arctan * arccot is_differentiable_in x )
assume A5: x in Z ; ::_thesis: arctan * arccot is_differentiable_in x
then A6: ( arccot . x > - 1 & arccot . x < 1 ) by A3;
 arccot is_differentiable_on Z by A2, SIN_COS9:82;
then  arccot is_differentiable_in x by A5, FDIFF_1:9;
hence  arctan * arccot is_differentiable_in x by A6, SIN_COS9:85; ::_thesis: verum
end;
then A7: arctan * arccot is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((arctan * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies ((arctan * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) )
assume A8: x in Z ; ::_thesis: ((arctan * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))))
then A9: ( arccot . x > - 1 & arccot . x < 1 ) by A3;
A10: arccot is_differentiable_on Z by A2, SIN_COS9:82;
then A11: arccot is_differentiable_in x by A8, FDIFF_1:9;
((arctan * arccot) `| Z) . x = diff ((arctan * arccot),x) by A7, A8, FDIFF_1:def_7
.= (diff (arccot,x)) / (1 + ((arccot . x) ^2)) by A11, A9, SIN_COS9:85
.= ((arccot `| Z) . x) / (1 + ((arccot . x) ^2)) by A8, A10, FDIFF_1:def_7
.= (- (1 / (1 + (x ^2)))) / (1 + ((arccot . x) ^2)) by A2, A8, SIN_COS9:82
.= - ((1 / (1 + (x ^2))) / (1 + ((arccot . x) ^2)))
.= - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) by XCMPLX_1:78 ;
hence  ((arctan * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) ; ::_thesis: verum
end;
hence  ( arctan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * arccot) `| Z) . x = - (1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:12
for Z being open Subset of REAL st Z c= dom (arccot * arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arccot . x > - 1 & arccot . x < 1 ) ) holds
( arccot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * arccot) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (arccot * arccot) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arccot . x > - 1 & arccot . x < 1 ) ) implies ( arccot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * arccot) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) ) ) )
assume that
A1: Z c= dom (arccot * arccot) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
( arccot . x > - 1 & arccot . x < 1 ) ; ::_thesis: ( arccot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * arccot) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) ) )
A4: for x being Real st x in Z holds
arccot * arccot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies arccot * arccot is_differentiable_in x )
assume A5: x in Z ; ::_thesis: arccot * arccot is_differentiable_in x
then A6: ( arccot . x > - 1 & arccot . x < 1 ) by A3;
 arccot is_differentiable_on Z by A2, SIN_COS9:82;
then  arccot is_differentiable_in x by A5, FDIFF_1:9;
hence  arccot * arccot is_differentiable_in x by A6, SIN_COS9:86; ::_thesis: verum
end;
then A7: arccot * arccot is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((arccot * arccot) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((arccot * arccot) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) )
assume A8: x in Z ; ::_thesis: ((arccot * arccot) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2)))
then A9: ( arccot . x > - 1 & arccot . x < 1 ) by A3;
A10: arccot is_differentiable_on Z by A2, SIN_COS9:82;
then A11: arccot is_differentiable_in x by A8, FDIFF_1:9;
((arccot * arccot) `| Z) . x = diff ((arccot * arccot),x) by A7, A8, FDIFF_1:def_7
.= - ((diff (arccot,x)) / (1 + ((arccot . x) ^2))) by A11, A9, SIN_COS9:86
.= - (((arccot `| Z) . x) / (1 + ((arccot . x) ^2))) by A8, A10, FDIFF_1:def_7
.= - ((- (1 / (1 + (x ^2)))) / (1 + ((arccot . x) ^2))) by A2, A8, SIN_COS9:82
.= (1 / (1 + (x ^2))) / (1 + ((arccot . x) ^2))
.= 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) by XCMPLX_1:78 ;
hence  ((arccot * arccot) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) ; ::_thesis: verum
end;
hence  ( arccot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * arccot) `| Z) . x = 1 / ((1 + (x ^2)) * (1 + ((arccot . x) ^2))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:13
for Z being open Subset of REAL st Z c= dom (sin * arctan) & Z c= ].(- 1),1.[ holds
( sin * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * arctan) `| Z) . x = (cos . (arctan . x)) / (1 + (x ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sin * arctan) & Z c= ].(- 1),1.[ implies ( sin * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * arctan) `| Z) . x = (cos . (arctan . x)) / (1 + (x ^2)) ) ) )
assume that
A1: Z c= dom (sin * arctan) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sin * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * arctan) `| Z) . x = (cos . (arctan . x)) / (1 + (x ^2)) ) )
A3: for x being Real st x in Z holds
sin * arctan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies sin * arctan is_differentiable_in x )
assume A4: x in Z ; ::_thesis: sin * arctan is_differentiable_in x
A5: sin is_differentiable_in arctan . x by SIN_COS:64;
 arctan is_differentiable_on Z by A2, SIN_COS9:81;
then  arctan is_differentiable_in x by A4, FDIFF_1:9;
hence  sin * arctan is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum
end;
then A6: sin * arctan is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((sin * arctan) `| Z) . x = (cos . (arctan . x)) / (1 + (x ^2))
proof
let x be Real; ::_thesis: ( x in Z implies ((sin * arctan) `| Z) . x = (cos . (arctan . x)) / (1 + (x ^2)) )
assume A7: x in Z ; ::_thesis: ((sin * arctan) `| Z) . x = (cos . (arctan . x)) / (1 + (x ^2))
A8: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A9: arctan is_differentiable_in x by A7, FDIFF_1:9;
A10: sin is_differentiable_in arctan . x by SIN_COS:64;
((sin * arctan) `| Z) . x = diff ((sin * arctan),x) by A6, A7, FDIFF_1:def_7
.= (diff (sin,(arctan . x))) * (diff (arctan,x)) by A9, A10, FDIFF_2:13
.= (cos . (arctan . x)) * (diff (arctan,x)) by SIN_COS:64
.= (cos . (arctan . x)) * ((arctan `| Z) . x) by A7, A8, FDIFF_1:def_7
.= (cos . (arctan . x)) * (1 / (1 + (x ^2))) by A2, A7, SIN_COS9:81
.= (cos . (arctan . x)) / (1 + (x ^2)) ;
hence  ((sin * arctan) `| Z) . x = (cos . (arctan . x)) / (1 + (x ^2)) ; ::_thesis: verum
end;
hence  ( sin * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * arctan) `| Z) . x = (cos . (arctan . x)) / (1 + (x ^2)) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:14
for Z being open Subset of REAL st Z c= dom (sin * arccot) & Z c= ].(- 1),1.[ holds
( sin * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sin * arccot) & Z c= ].(- 1),1.[ implies ( sin * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (sin * arccot) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sin * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) ) )
A3: for x being Real st x in Z holds
sin * arccot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies sin * arccot is_differentiable_in x )
assume A4: x in Z ; ::_thesis: sin * arccot is_differentiable_in x
A5: sin is_differentiable_in arccot . x by SIN_COS:64;
 arccot is_differentiable_on Z by A2, SIN_COS9:82;
then  arccot is_differentiable_in x by A4, FDIFF_1:9;
hence  sin * arccot is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum
end;
then A6: sin * arccot is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) )
assume A7: x in Z ; ::_thesis: ((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2)))
A8: arccot is_differentiable_on Z by A2, SIN_COS9:82;
then A9: arccot is_differentiable_in x by A7, FDIFF_1:9;
A10: sin is_differentiable_in arccot . x by SIN_COS:64;
((sin * arccot) `| Z) . x = diff ((sin * arccot),x) by A6, A7, FDIFF_1:def_7
.= (diff (sin,(arccot . x))) * (diff (arccot,x)) by A9, A10, FDIFF_2:13
.= (cos . (arccot . x)) * (diff (arccot,x)) by SIN_COS:64
.= (cos . (arccot . x)) * ((arccot `| Z) . x) by A7, A8, FDIFF_1:def_7
.= (cos . (arccot . x)) * (- (1 / (1 + (x ^2)))) by A2, A7, SIN_COS9:82
.= - ((cos . (arccot . x)) / (1 + (x ^2))) ;
hence  ((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( sin * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * arccot) `| Z) . x = - ((cos . (arccot . x)) / (1 + (x ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:15
for Z being open Subset of REAL st Z c= dom (cos * arctan) & Z c= ].(- 1),1.[ holds
( cos * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cos * arctan) & Z c= ].(- 1),1.[ implies ( cos * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (cos * arctan) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cos * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2))) ) )
A3: for x being Real st x in Z holds
cos * arctan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cos * arctan is_differentiable_in x )
assume A4: x in Z ; ::_thesis: cos * arctan is_differentiable_in x
A5: cos is_differentiable_in arctan . x by SIN_COS:63;
 arctan is_differentiable_on Z by A2, SIN_COS9:81;
then  arctan is_differentiable_in x by A4, FDIFF_1:9;
hence  cos * arctan is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum
end;
then A6: cos * arctan is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2))) )
assume A7: x in Z ; ::_thesis: ((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2)))
A8: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A9: arctan is_differentiable_in x by A7, FDIFF_1:9;
A10: cos is_differentiable_in arctan . x by SIN_COS:63;
((cos * arctan) `| Z) . x = diff ((cos * arctan),x) by A6, A7, FDIFF_1:def_7
.= (diff (cos,(arctan . x))) * (diff (arctan,x)) by A9, A10, FDIFF_2:13
.= (- (sin . (arctan . x))) * (diff (arctan,x)) by SIN_COS:63
.= - ((sin . (arctan . x)) * (diff (arctan,x)))
.= - ((sin . (arctan . x)) * ((arctan `| Z) . x)) by A7, A8, FDIFF_1:def_7
.= - ((sin . (arctan . x)) * (1 / (1 + (x ^2)))) by A2, A7, SIN_COS9:81
.= - ((sin . (arctan . x)) / (1 + (x ^2))) ;
hence  ((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( cos * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * arctan) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:16
for Z being open Subset of REAL st Z c= dom (cos * arccot) & Z c= ].(- 1),1.[ holds
( cos * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * arccot) `| Z) . x = (sin . (arccot . x)) / (1 + (x ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cos * arccot) & Z c= ].(- 1),1.[ implies ( cos * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * arccot) `| Z) . x = (sin . (arccot . x)) / (1 + (x ^2)) ) ) )
assume that
A1: Z c= dom (cos * arccot) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cos * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * arccot) `| Z) . x = (sin . (arccot . x)) / (1 + (x ^2)) ) )
A3: for x being Real st x in Z holds
cos * arccot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cos * arccot is_differentiable_in x )
assume A4: x in Z ; ::_thesis: cos * arccot is_differentiable_in x
A5: cos is_differentiable_in arccot . x by SIN_COS:63;
 arccot is_differentiable_on Z by A2, SIN_COS9:82;
then  arccot is_differentiable_in x by A4, FDIFF_1:9;
hence  cos * arccot is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum
end;
then A6: cos * arccot is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((cos * arccot) `| Z) . x = (sin . (arccot . x)) / (1 + (x ^2))
proof
let x be Real; ::_thesis: ( x in Z implies ((cos * arccot) `| Z) . x = (sin . (arccot . x)) / (1 + (x ^2)) )
assume A7: x in Z ; ::_thesis: ((cos * arccot) `| Z) . x = (sin . (arccot . x)) / (1 + (x ^2))
A8: arccot is_differentiable_on Z by A2, SIN_COS9:82;
then A9: arccot is_differentiable_in x by A7, FDIFF_1:9;
A10: cos is_differentiable_in arccot . x by SIN_COS:63;
((cos * arccot) `| Z) . x = diff ((cos * arccot),x) by A6, A7, FDIFF_1:def_7
.= (diff (cos,(arccot . x))) * (diff (arccot,x)) by A9, A10, FDIFF_2:13
.= (- (sin . (arccot . x))) * (diff (arccot,x)) by SIN_COS:63
.= - ((sin . (arccot . x)) * (diff (arccot,x)))
.= - ((sin . (arccot . x)) * ((arccot `| Z) . x)) by A7, A8, FDIFF_1:def_7
.= - ((sin . (arccot . x)) * (- (1 / (1 + (x ^2))))) by A2, A7, SIN_COS9:82
.= (sin . (arccot . x)) / (1 + (x ^2)) ;
hence  ((cos * arccot) `| Z) . x = (sin . (arccot . x)) / (1 + (x ^2)) ; ::_thesis: verum
end;
hence  ( cos * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * arccot) `| Z) . x = (sin . (arccot . x)) / (1 + (x ^2)) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:17
for Z being open Subset of REAL st Z c= dom (tan * arctan) & Z c= ].(- 1),1.[ holds
( tan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * arctan) `| Z) . x = 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (tan * arctan) & Z c= ].(- 1),1.[ implies ( tan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * arctan) `| Z) . x = 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (tan * arctan) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( tan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * arctan) `| Z) . x = 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) )
A3: for x being Real st x in Z holds
tan * arctan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies tan * arctan is_differentiable_in x )
assume A4: x in Z ; ::_thesis: tan * arctan is_differentiable_in x
then  arctan . x in dom tan by A1, FUNCT_1:11;
then  cos . (arctan . x) <> 0 by FDIFF_8:1;
then A5: tan is_differentiable_in arctan . x by FDIFF_7:46;
 arctan is_differentiable_on Z by A2, SIN_COS9:81;
then  arctan is_differentiable_in x by A4, FDIFF_1:9;
hence  tan * arctan is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum
end;
then A6: tan * arctan is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((tan * arctan) `| Z) . x = 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((tan * arctan) `| Z) . x = 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) )
assume A7: x in Z ; ::_thesis: ((tan * arctan) `| Z) . x = 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2)))
then  arctan . x in dom tan by A1, FUNCT_1:11;
then A8: cos . (arctan . x) <> 0 by FDIFF_8:1;
then A9: tan is_differentiable_in arctan . x by FDIFF_7:46;
A10: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A11: arctan is_differentiable_in x by A7, FDIFF_1:9;
((tan * arctan) `| Z) . x = diff ((tan * arctan),x) by A6, A7, FDIFF_1:def_7
.= (diff (tan,(arctan . x))) * (diff (arctan,x)) by A11, A9, FDIFF_2:13
.= (1 / ((cos . (arctan . x)) ^2)) * (diff (arctan,x)) by A8, FDIFF_7:46
.= (1 / ((cos . (arctan . x)) ^2)) * ((arctan `| Z) . x) by A7, A10, FDIFF_1:def_7
.= (1 / ((cos . (arctan . x)) ^2)) * (1 / (1 + (x ^2))) by A2, A7, SIN_COS9:81
.= 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) by XCMPLX_1:102 ;
hence  ((tan * arctan) `| Z) . x = 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( tan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * arctan) `| Z) . x = 1 / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:18
for Z being open Subset of REAL st Z c= dom (tan * arccot) & Z c= ].(- 1),1.[ holds
( tan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (tan * arccot) & Z c= ].(- 1),1.[ implies ( tan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) ) )
assume that
A1: Z c= dom (tan * arccot) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( tan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) )
A3: for x being Real st x in Z holds
tan * arccot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies tan * arccot is_differentiable_in x )
assume A4: x in Z ; ::_thesis: tan * arccot is_differentiable_in x
then  arccot . x in dom tan by A1, FUNCT_1:11;
then  cos . (arccot . x) <> 0 by FDIFF_8:1;
then A5: tan is_differentiable_in arccot . x by FDIFF_7:46;
 arccot is_differentiable_on Z by A2, SIN_COS9:82;
then  arccot is_differentiable_in x by A4, FDIFF_1:9;
hence  tan * arccot is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum
end;
then A6: tan * arccot is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies ((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) )
assume A7: x in Z ; ::_thesis: ((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2))))
then  arccot . x in dom tan by A1, FUNCT_1:11;
then A8: cos . (arccot . x) <> 0 by FDIFF_8:1;
then A9: tan is_differentiable_in arccot . x by FDIFF_7:46;
A10: arccot is_differentiable_on Z by A2, SIN_COS9:82;
then A11: arccot is_differentiable_in x by A7, FDIFF_1:9;
((tan * arccot) `| Z) . x = diff ((tan * arccot),x) by A6, A7, FDIFF_1:def_7
.= (diff (tan,(arccot . x))) * (diff (arccot,x)) by A11, A9, FDIFF_2:13
.= (1 / ((cos . (arccot . x)) ^2)) * (diff (arccot,x)) by A8, FDIFF_7:46
.= (1 / ((cos . (arccot . x)) ^2)) * ((arccot `| Z) . x) by A7, A10, FDIFF_1:def_7
.= (1 / ((cos . (arccot . x)) ^2)) * (- (1 / (1 + (x ^2)))) by A2, A7, SIN_COS9:82
.= - ((1 / ((cos . (arccot . x)) ^2)) * (1 / (1 + (x ^2))))
.= - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) by XCMPLX_1:102 ;
hence  ((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ; ::_thesis: verum
end;
hence  ( tan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:19
for Z being open Subset of REAL st Z c= dom (cot * arctan) & Z c= ].(- 1),1.[ holds
( cot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cot * arctan) & Z c= ].(- 1),1.[ implies ( cot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) ) )
assume that
A1: Z c= dom (cot * arctan) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) )
A3: for x being Real st x in Z holds
cot * arctan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cot * arctan is_differentiable_in x )
assume A4: x in Z ; ::_thesis: cot * arctan is_differentiable_in x
then  arctan . x in dom cot by A1, FUNCT_1:11;
then  sin . (arctan . x) <> 0 by FDIFF_8:2;
then A5: cot is_differentiable_in arctan . x by FDIFF_7:47;
 arctan is_differentiable_on Z by A2, SIN_COS9:81;
then  arctan is_differentiable_in x by A4, FDIFF_1:9;
hence  cot * arctan is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum
end;
then A6: cot * arctan is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies ((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) )
assume A7: x in Z ; ::_thesis: ((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2))))
then  arctan . x in dom cot by A1, FUNCT_1:11;
then A8: sin . (arctan . x) <> 0 by FDIFF_8:2;
then A9: cot is_differentiable_in arctan . x by FDIFF_7:47;
A10: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A11: arctan is_differentiable_in x by A7, FDIFF_1:9;
((cot * arctan) `| Z) . x = diff ((cot * arctan),x) by A6, A7, FDIFF_1:def_7
.= (diff (cot,(arctan . x))) * (diff (arctan,x)) by A11, A9, FDIFF_2:13
.= (- (1 / ((sin . (arctan . x)) ^2))) * (diff (arctan,x)) by A8, FDIFF_7:47
.= - ((1 / ((sin . (arctan . x)) ^2)) * (diff (arctan,x)))
.= - ((1 / ((sin . (arctan . x)) ^2)) * ((arctan `| Z) . x)) by A7, A10, FDIFF_1:def_7
.= - ((1 / ((sin . (arctan . x)) ^2)) * (1 / (1 + (x ^2)))) by A2, A7, SIN_COS9:81
.= - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) by XCMPLX_1:102 ;
hence  ((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ; ::_thesis: verum
end;
hence  ( cot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:20
for Z being open Subset of REAL st Z c= dom (cot * arccot) & Z c= ].(- 1),1.[ holds
( cot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * arccot) `| Z) . x = 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cot * arccot) & Z c= ].(- 1),1.[ implies ( cot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * arccot) `| Z) . x = 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (cot * arccot) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * arccot) `| Z) . x = 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) )
A3: for x being Real st x in Z holds
cot * arccot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cot * arccot is_differentiable_in x )
assume A4: x in Z ; ::_thesis: cot * arccot is_differentiable_in x
then  arccot . x in dom cot by A1, FUNCT_1:11;
then  sin . (arccot . x) <> 0 by FDIFF_8:2;
then A5: cot is_differentiable_in arccot . x by FDIFF_7:47;
 arccot is_differentiable_on Z by A2, SIN_COS9:82;
then  arccot is_differentiable_in x by A4, FDIFF_1:9;
hence  cot * arccot is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum
end;
then A6: cot * arccot is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((cot * arccot) `| Z) . x = 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((cot * arccot) `| Z) . x = 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) )
assume A7: x in Z ; ::_thesis: ((cot * arccot) `| Z) . x = 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2)))
then  arccot . x in dom cot by A1, FUNCT_1:11;
then A8: sin . (arccot . x) <> 0 by FDIFF_8:2;
then A9: cot is_differentiable_in arccot . x by FDIFF_7:47;
A10: arccot is_differentiable_on Z by A2, SIN_COS9:82;
then A11: arccot is_differentiable_in x by A7, FDIFF_1:9;
((cot * arccot) `| Z) . x = diff ((cot * arccot),x) by A6, A7, FDIFF_1:def_7
.= (diff (cot,(arccot . x))) * (diff (arccot,x)) by A11, A9, FDIFF_2:13
.= (- (1 / ((sin . (arccot . x)) ^2))) * (diff (arccot,x)) by A8, FDIFF_7:47
.= - ((1 / ((sin . (arccot . x)) ^2)) * (diff (arccot,x)))
.= - ((1 / ((sin . (arccot . x)) ^2)) * ((arccot `| Z) . x)) by A7, A10, FDIFF_1:def_7
.= - ((1 / ((sin . (arccot . x)) ^2)) * (- (1 / (1 + (x ^2))))) by A2, A7, SIN_COS9:82
.= (1 / ((sin . (arccot . x)) ^2)) * (1 / (1 + (x ^2)))
.= 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) by XCMPLX_1:102 ;
hence  ((cot * arccot) `| Z) . x = 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( cot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * arccot) `| Z) . x = 1 / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) ) by A1, A3, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:21
for Z being open Subset of REAL st Z c= dom (sec * arctan) & Z c= ].(- 1),1.[ holds
( sec * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * arctan) `| Z) . x = (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sec * arctan) & Z c= ].(- 1),1.[ implies ( sec * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * arctan) `| Z) . x = (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (sec * arctan) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sec * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * arctan) `| Z) . x = (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) )
A3: for x being Real st x in Z holds
cos . (arctan . x) <> 0
proof
let x be Real; ::_thesis: ( x in Z implies cos . (arctan . x) <> 0 )
assume  x in Z ; ::_thesis: cos . (arctan . x) <> 0
then  arctan . x in dom sec by A1, FUNCT_1:11;
hence  cos . (arctan . x) <> 0 by RFUNCT_1:3; ::_thesis: verum
end;
A4: for x being Real st x in Z holds
sec * arctan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies sec * arctan is_differentiable_in x )
assume A5: x in Z ; ::_thesis: sec * arctan is_differentiable_in x
then  cos . (arctan . x) <> 0 by A3;
then A6: sec is_differentiable_in arctan . x by FDIFF_9:1;
 arctan is_differentiable_on Z by A2, SIN_COS9:81;
then  arctan is_differentiable_in x by A5, FDIFF_1:9;
hence  sec * arctan is_differentiable_in x by A6, FDIFF_2:13; ::_thesis: verum
end;
then A7: sec * arctan is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((sec * arctan) `| Z) . x = (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((sec * arctan) `| Z) . x = (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) )
assume A8: x in Z ; ::_thesis: ((sec * arctan) `| Z) . x = (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2)))
then A9: cos . (arctan . x) <> 0 by A3;
 cos . (arctan . x) <> 0 by A3, A8;
then A10: sec is_differentiable_in arctan . x by FDIFF_9:1;
A11: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A12: arctan is_differentiable_in x by A8, FDIFF_1:9;
((sec * arctan) `| Z) . x = diff ((sec * arctan),x) by A7, A8, FDIFF_1:def_7
.= (diff (sec,(arctan . x))) * (diff (arctan,x)) by A12, A10, FDIFF_2:13
.= ((sin . (arctan . x)) / ((cos . (arctan . x)) ^2)) * (diff (arctan,x)) by A9, FDIFF_9:1
.= ((sin . (arctan . x)) / ((cos . (arctan . x)) ^2)) * ((arctan `| Z) . x) by A8, A11, FDIFF_1:def_7
.= ((sin . (arctan . x)) / ((cos . (arctan . x)) ^2)) * (1 / (1 + (x ^2))) by A2, A8, SIN_COS9:81
.= ((sin . (arctan . x)) * 1) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) by XCMPLX_1:76
.= (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ;
hence  ((sec * arctan) `| Z) . x = (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( sec * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * arctan) `| Z) . x = (sin . (arctan . x)) / (((cos . (arctan . x)) ^2) * (1 + (x ^2))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:22
for Z being open Subset of REAL st Z c= dom (sec * arccot) & Z c= ].(- 1),1.[ holds
( sec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * arccot) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sec * arccot) & Z c= ].(- 1),1.[ implies ( sec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * arccot) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) ) )
assume that
A1: Z c= dom (sec * arccot) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * arccot) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) )
A3: for x being Real st x in Z holds
cos . (arccot . x) <> 0
proof
let x be Real; ::_thesis: ( x in Z implies cos . (arccot . x) <> 0 )
assume  x in Z ; ::_thesis: cos . (arccot . x) <> 0
then  arccot . x in dom sec by A1, FUNCT_1:11;
hence  cos . (arccot . x) <> 0 by RFUNCT_1:3; ::_thesis: verum
end;
A4: for x being Real st x in Z holds
sec * arccot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies sec * arccot is_differentiable_in x )
assume A5: x in Z ; ::_thesis: sec * arccot is_differentiable_in x
then  cos . (arccot . x) <> 0 by A3;
then A6: sec is_differentiable_in arccot . x by FDIFF_9:1;
 arccot is_differentiable_on Z by A2, SIN_COS9:82;
then  arccot is_differentiable_in x by A5, FDIFF_1:9;
hence  sec * arccot is_differentiable_in x by A6, FDIFF_2:13; ::_thesis: verum
end;
then A7: sec * arccot is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((sec * arccot) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies ((sec * arccot) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) )
assume A8: x in Z ; ::_thesis: ((sec * arccot) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2))))
then A9: cos . (arccot . x) <> 0 by A3;
 cos . (arccot . x) <> 0 by A3, A8;
then A10: sec is_differentiable_in arccot . x by FDIFF_9:1;
A11: arccot is_differentiable_on Z by A2, SIN_COS9:82;
then A12: arccot is_differentiable_in x by A8, FDIFF_1:9;
((sec * arccot) `| Z) . x = diff ((sec * arccot),x) by A7, A8, FDIFF_1:def_7
.= (diff (sec,(arccot . x))) * (diff (arccot,x)) by A12, A10, FDIFF_2:13
.= ((sin . (arccot . x)) / ((cos . (arccot . x)) ^2)) * (diff (arccot,x)) by A9, FDIFF_9:1
.= ((sin . (arccot . x)) / ((cos . (arccot . x)) ^2)) * ((arccot `| Z) . x) by A8, A11, FDIFF_1:def_7
.= ((sin . (arccot . x)) / ((cos . (arccot . x)) ^2)) * (- (1 / (1 + (x ^2)))) by A2, A8, SIN_COS9:82
.= - (((sin . (arccot . x)) / ((cos . (arccot . x)) ^2)) * (1 / (1 + (x ^2))))
.= - (((sin . (arccot . x)) * 1) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) by XCMPLX_1:76
.= - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ;
hence  ((sec * arccot) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ; ::_thesis: verum
end;
hence  ( sec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * arccot) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:23
for Z being open Subset of REAL st Z c= dom (cosec * arctan) & Z c= ].(- 1),1.[ holds
( cosec * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * arctan) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cosec * arctan) & Z c= ].(- 1),1.[ implies ( cosec * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * arctan) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) ) )
assume that
A1: Z c= dom (cosec * arctan) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cosec * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * arctan) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) )
A3: for x being Real st x in Z holds
sin . (arctan . x) <> 0
proof
let x be Real; ::_thesis: ( x in Z implies sin . (arctan . x) <> 0 )
assume  x in Z ; ::_thesis: sin . (arctan . x) <> 0
then  arctan . x in dom cosec by A1, FUNCT_1:11;
hence  sin . (arctan . x) <> 0 by RFUNCT_1:3; ::_thesis: verum
end;
A4: for x being Real st x in Z holds
cosec * arctan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cosec * arctan is_differentiable_in x )
assume A5: x in Z ; ::_thesis: cosec * arctan is_differentiable_in x
then  sin . (arctan . x) <> 0 by A3;
then A6: cosec is_differentiable_in arctan . x by FDIFF_9:2;
 arctan is_differentiable_on Z by A2, SIN_COS9:81;
then  arctan is_differentiable_in x by A5, FDIFF_1:9;
hence  cosec * arctan is_differentiable_in x by A6, FDIFF_2:13; ::_thesis: verum
end;
then A7: cosec * arctan is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((cosec * arctan) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies ((cosec * arctan) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) )
assume A8: x in Z ; ::_thesis: ((cosec * arctan) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2))))
then A9: sin . (arctan . x) <> 0 by A3;
 sin . (arctan . x) <> 0 by A3, A8;
then A10: cosec is_differentiable_in arctan . x by FDIFF_9:2;
A11: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A12: arctan is_differentiable_in x by A8, FDIFF_1:9;
((cosec * arctan) `| Z) . x = diff ((cosec * arctan),x) by A7, A8, FDIFF_1:def_7
.= (diff (cosec,(arctan . x))) * (diff (arctan,x)) by A12, A10, FDIFF_2:13
.= (- ((cos . (arctan . x)) / ((sin . (arctan . x)) ^2))) * (diff (arctan,x)) by A9, FDIFF_9:2
.= - (((cos . (arctan . x)) / ((sin . (arctan . x)) ^2)) * (diff (arctan,x)))
.= - (((cos . (arctan . x)) / ((sin . (arctan . x)) ^2)) * ((arctan `| Z) . x)) by A8, A11, FDIFF_1:def_7
.= - (((cos . (arctan . x)) / ((sin . (arctan . x)) ^2)) * (1 / (1 + (x ^2)))) by A2, A8, SIN_COS9:81
.= - (((cos . (arctan . x)) * 1) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) by XCMPLX_1:76
.= - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ;
hence  ((cosec * arctan) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ; ::_thesis: verum
end;
hence  ( cosec * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * arctan) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:24
for Z being open Subset of REAL st Z c= dom (cosec * arccot) & Z c= ].(- 1),1.[ holds
( cosec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * arccot) `| Z) . x = (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cosec * arccot) & Z c= ].(- 1),1.[ implies ( cosec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * arccot) `| Z) . x = (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (cosec * arccot) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cosec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * arccot) `| Z) . x = (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) )
A3: for x being Real st x in Z holds
sin . (arccot . x) <> 0
proof
let x be Real; ::_thesis: ( x in Z implies sin . (arccot . x) <> 0 )
assume  x in Z ; ::_thesis: sin . (arccot . x) <> 0
then  arccot . x in dom cosec by A1, FUNCT_1:11;
hence  sin . (arccot . x) <> 0 by RFUNCT_1:3; ::_thesis: verum
end;
A4: for x being Real st x in Z holds
cosec * arccot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cosec * arccot is_differentiable_in x )
assume A5: x in Z ; ::_thesis: cosec * arccot is_differentiable_in x
then  sin . (arccot . x) <> 0 by A3;
then A6: cosec is_differentiable_in arccot . x by FDIFF_9:2;
 arccot is_differentiable_on Z by A2, SIN_COS9:82;
then  arccot is_differentiable_in x by A5, FDIFF_1:9;
hence  cosec * arccot is_differentiable_in x by A6, FDIFF_2:13; ::_thesis: verum
end;
then A7: cosec * arccot is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((cosec * arccot) `| Z) . x = (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((cosec * arccot) `| Z) . x = (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) )
assume A8: x in Z ; ::_thesis: ((cosec * arccot) `| Z) . x = (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2)))
then A9: sin . (arccot . x) <> 0 by A3;
 sin . (arccot . x) <> 0 by A3, A8;
then A10: cosec is_differentiable_in arccot . x by FDIFF_9:2;
A11: arccot is_differentiable_on Z by A2, SIN_COS9:82;
then A12: arccot is_differentiable_in x by A8, FDIFF_1:9;
((cosec * arccot) `| Z) . x = diff ((cosec * arccot),x) by A7, A8, FDIFF_1:def_7
.= (diff (cosec,(arccot . x))) * (diff (arccot,x)) by A12, A10, FDIFF_2:13
.= (- ((cos . (arccot . x)) / ((sin . (arccot . x)) ^2))) * (diff (arccot,x)) by A9, FDIFF_9:2
.= - (((cos . (arccot . x)) / ((sin . (arccot . x)) ^2)) * (diff (arccot,x)))
.= - (((cos . (arccot . x)) / ((sin . (arccot . x)) ^2)) * ((arccot `| Z) . x)) by A8, A11, FDIFF_1:def_7
.= - (((cos . (arccot . x)) / ((sin . (arccot . x)) ^2)) * (- (1 / (1 + (x ^2))))) by A2, A8, SIN_COS9:82
.= ((cos . (arccot . x)) / ((sin . (arccot . x)) ^2)) * (1 / (1 + (x ^2)))
.= ((cos . (arccot . x)) * 1) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) by XCMPLX_1:76
.= (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ;
hence  ((cosec * arccot) `| Z) . x = (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( cosec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * arccot) `| Z) . x = (cos . (arccot . x)) / (((sin . (arccot . x)) ^2) * (1 + (x ^2))) ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:25
for Z being open Subset of REAL st Z c= dom (sin (#) arctan) & Z c= ].(- 1),1.[ holds
( sin (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sin (#) arctan) & Z c= ].(- 1),1.[ implies ( sin (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (sin (#) arctan) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sin (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ) )
A3: arctan is_differentiable_on Z by A2, SIN_COS9:81;
A4: for x being Real st x in Z holds
sin is_differentiable_in x by SIN_COS:64;
 Z c= (dom sin) /\ (dom arctan) by A1, VALUED_1:def_4;
then  Z c= dom sin by XBOOLE_1:18;
then A5: sin is_differentiable_on Z by A4, FDIFF_1:9;
 for x being Real st x in Z holds
((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) )
assume A6: x in Z ; ::_thesis: ((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2)))
then ((sin (#) arctan) `| Z) . x = ((arctan . x) * (diff (sin,x))) + ((sin . x) * (diff (arctan,x))) by A1, A5, A3, FDIFF_1:21
.= ((arctan . x) * (cos . x)) + ((sin . x) * (diff (arctan,x))) by SIN_COS:64
.= ((cos . x) * (arctan . x)) + ((sin . x) * ((arctan `| Z) . x)) by A3, A6, FDIFF_1:def_7
.= ((cos . x) * (arctan . x)) + ((sin . x) * (1 / (1 + (x ^2)))) by A2, A6, SIN_COS9:81
.= ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ;
hence  ((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( sin (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:26
for Z being open Subset of REAL st Z c= dom (sin (#) arccot) & Z c= ].(- 1),1.[ holds
( sin (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) arccot) `| Z) . x = ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sin (#) arccot) & Z c= ].(- 1),1.[ implies ( sin (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) arccot) `| Z) . x = ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (sin (#) arccot) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sin (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) arccot) `| Z) . x = ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) ) )
A3: arccot is_differentiable_on Z by A2, SIN_COS9:82;
A4: for x being Real st x in Z holds
sin is_differentiable_in x by SIN_COS:64;
 Z c= (dom sin) /\ (dom arccot) by A1, VALUED_1:def_4;
then  Z c= dom sin by XBOOLE_1:18;
then A5: sin is_differentiable_on Z by A4, FDIFF_1:9;
 for x being Real st x in Z holds
((sin (#) arccot) `| Z) . x = ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((sin (#) arccot) `| Z) . x = ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) )
assume A6: x in Z ; ::_thesis: ((sin (#) arccot) `| Z) . x = ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2)))
then ((sin (#) arccot) `| Z) . x = ((arccot . x) * (diff (sin,x))) + ((sin . x) * (diff (arccot,x))) by A1, A5, A3, FDIFF_1:21
.= ((arccot . x) * (cos . x)) + ((sin . x) * (diff (arccot,x))) by SIN_COS:64
.= ((cos . x) * (arccot . x)) + ((sin . x) * ((arccot `| Z) . x)) by A3, A6, FDIFF_1:def_7
.= ((cos . x) * (arccot . x)) + ((sin . x) * (- (1 / (1 + (x ^2))))) by A2, A6, SIN_COS9:82
.= ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) ;
hence  ((sin (#) arccot) `| Z) . x = ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( sin (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) arccot) `| Z) . x = ((cos . x) * (arccot . x)) - ((sin . x) / (1 + (x ^2))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:27
for Z being open Subset of REAL st Z c= dom (cos (#) arctan) & Z c= ].(- 1),1.[ holds
( cos (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) arctan) `| Z) . x = (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cos (#) arctan) & Z c= ].(- 1),1.[ implies ( cos (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) arctan) `| Z) . x = (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (cos (#) arctan) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cos (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) arctan) `| Z) . x = (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) ) )
A3: arctan is_differentiable_on Z by A2, SIN_COS9:81;
A4: for x being Real st x in Z holds
cos is_differentiable_in x by SIN_COS:63;
 Z c= (dom cos) /\ (dom arctan) by A1, VALUED_1:def_4;
then  Z c= dom cos by XBOOLE_1:18;
then A5: cos is_differentiable_on Z by A4, FDIFF_1:9;
 for x being Real st x in Z holds
((cos (#) arctan) `| Z) . x = (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((cos (#) arctan) `| Z) . x = (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) )
assume A6: x in Z ; ::_thesis: ((cos (#) arctan) `| Z) . x = (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2)))
then ((cos (#) arctan) `| Z) . x = ((arctan . x) * (diff (cos,x))) + ((cos . x) * (diff (arctan,x))) by A1, A5, A3, FDIFF_1:21
.= ((arctan . x) * (- (sin . x))) + ((cos . x) * (diff (arctan,x))) by SIN_COS:63
.= (- ((sin . x) * (arctan . x))) + ((cos . x) * ((arctan `| Z) . x)) by A3, A6, FDIFF_1:def_7
.= (- ((sin . x) * (arctan . x))) + ((cos . x) * (1 / (1 + (x ^2)))) by A2, A6, SIN_COS9:81
.= (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) ;
hence  ((cos (#) arctan) `| Z) . x = (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( cos (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) arctan) `| Z) . x = (- ((sin . x) * (arctan . x))) + ((cos . x) / (1 + (x ^2))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:28
for Z being open Subset of REAL st Z c= dom (cos (#) arccot) & Z c= ].(- 1),1.[ holds
( cos (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) arccot) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cos (#) arccot) & Z c= ].(- 1),1.[ implies ( cos (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) arccot) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (cos (#) arccot) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cos (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) arccot) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) ) )
A3: arccot is_differentiable_on Z by A2, SIN_COS9:82;
A4: for x being Real st x in Z holds
cos is_differentiable_in x by SIN_COS:63;
 Z c= (dom cos) /\ (dom arccot) by A1, VALUED_1:def_4;
then  Z c= dom cos by XBOOLE_1:18;
then A5: cos is_differentiable_on Z by A4, FDIFF_1:9;
 for x being Real st x in Z holds
((cos (#) arccot) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((cos (#) arccot) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) )
assume A6: x in Z ; ::_thesis: ((cos (#) arccot) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2)))
then ((cos (#) arccot) `| Z) . x = ((arccot . x) * (diff (cos,x))) + ((cos . x) * (diff (arccot,x))) by A1, A5, A3, FDIFF_1:21
.= ((arccot . x) * (- (sin . x))) + ((cos . x) * (diff (arccot,x))) by SIN_COS:63
.= (- ((sin . x) * (arccot . x))) + ((cos . x) * ((arccot `| Z) . x)) by A3, A6, FDIFF_1:def_7
.= (- ((sin . x) * (arccot . x))) + ((cos . x) * (- (1 / (1 + (x ^2))))) by A2, A6, SIN_COS9:82
.= (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) ;
hence  ((cos (#) arccot) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( cos (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) arccot) `| Z) . x = (- ((sin . x) * (arccot . x))) - ((cos . x) / (1 + (x ^2))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:29
for Z being open Subset of REAL st Z c= dom (tan (#) arctan) & Z c= ].(- 1),1.[ holds
( tan (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) arctan) `| Z) . x = ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (tan (#) arctan) & Z c= ].(- 1),1.[ implies ( tan (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) arctan) `| Z) . x = ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (tan (#) arctan) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( tan (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) arctan) `| Z) . x = ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) ) )
A3: arctan is_differentiable_on Z by A2, SIN_COS9:81;
 Z c= (dom tan) /\ (dom arctan) by A1, VALUED_1:def_4;
then A4: Z c= dom tan by XBOOLE_1:18;
 for x being Real st x in Z holds
tan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies tan is_differentiable_in x )
assume  x in Z ; ::_thesis: tan is_differentiable_in x
then  cos . x <> 0 by A4, FDIFF_8:1;
hence  tan is_differentiable_in x by FDIFF_7:46; ::_thesis: verum
end;
then A5: tan is_differentiable_on Z by A4, FDIFF_1:9;
 for x being Real st x in Z holds
((tan (#) arctan) `| Z) . x = ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((tan (#) arctan) `| Z) . x = ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) )
assume A6: x in Z ; ::_thesis: ((tan (#) arctan) `| Z) . x = ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2)))
then A7: cos . x <> 0 by A4, FDIFF_8:1;
((tan (#) arctan) `| Z) . x = ((arctan . x) * (diff (tan,x))) + ((tan . x) * (diff (arctan,x))) by A1, A5, A3, A6, FDIFF_1:21
.= ((arctan . x) * (1 / ((cos . x) ^2))) + ((tan . x) * (diff (arctan,x))) by A7, FDIFF_7:46
.= ((arctan . x) / ((cos . x) ^2)) + ((tan . x) * ((arctan `| Z) . x)) by A3, A6, FDIFF_1:def_7
.= ((arctan . x) / ((cos . x) ^2)) + ((tan . x) * (1 / (1 + (x ^2)))) by A2, A6, SIN_COS9:81
.= ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) ;
hence  ((tan (#) arctan) `| Z) . x = ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( tan (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) arctan) `| Z) . x = ((arctan . x) / ((cos . x) ^2)) + ((tan . x) / (1 + (x ^2))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:30
for Z being open Subset of REAL st Z c= dom (tan (#) arccot) & Z c= ].(- 1),1.[ holds
( tan (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) arccot) `| Z) . x = ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (tan (#) arccot) & Z c= ].(- 1),1.[ implies ( tan (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) arccot) `| Z) . x = ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (tan (#) arccot) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( tan (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) arccot) `| Z) . x = ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) ) )
A3: arccot is_differentiable_on Z by A2, SIN_COS9:82;
 Z c= (dom tan) /\ (dom arccot) by A1, VALUED_1:def_4;
then A4: Z c= dom tan by XBOOLE_1:18;
 for x being Real st x in Z holds
tan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies tan is_differentiable_in x )
assume  x in Z ; ::_thesis: tan is_differentiable_in x
then  cos . x <> 0 by A4, FDIFF_8:1;
hence  tan is_differentiable_in x by FDIFF_7:46; ::_thesis: verum
end;
then A5: tan is_differentiable_on Z by A4, FDIFF_1:9;
 for x being Real st x in Z holds
((tan (#) arccot) `| Z) . x = ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((tan (#) arccot) `| Z) . x = ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) )
assume A6: x in Z ; ::_thesis: ((tan (#) arccot) `| Z) . x = ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2)))
then A7: cos . x <> 0 by A4, FDIFF_8:1;
((tan (#) arccot) `| Z) . x = ((arccot . x) * (diff (tan,x))) + ((tan . x) * (diff (arccot,x))) by A1, A5, A3, A6, FDIFF_1:21
.= ((arccot . x) * (1 / ((cos . x) ^2))) + ((tan . x) * (diff (arccot,x))) by A7, FDIFF_7:46
.= ((arccot . x) / ((cos . x) ^2)) + ((tan . x) * ((arccot `| Z) . x)) by A3, A6, FDIFF_1:def_7
.= ((arccot . x) / ((cos . x) ^2)) + ((tan . x) * (- (1 / (1 + (x ^2))))) by A2, A6, SIN_COS9:82
.= ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) ;
hence  ((tan (#) arccot) `| Z) . x = ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( tan (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) arccot) `| Z) . x = ((arccot . x) / ((cos . x) ^2)) - ((tan . x) / (1 + (x ^2))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:31
for Z being open Subset of REAL st Z c= dom (cot (#) arctan) & Z c= ].(- 1),1.[ holds
( cot (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) arctan) `| Z) . x = (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cot (#) arctan) & Z c= ].(- 1),1.[ implies ( cot (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) arctan) `| Z) . x = (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (cot (#) arctan) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cot (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) arctan) `| Z) . x = (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) ) )
A3: arctan is_differentiable_on Z by A2, SIN_COS9:81;
 Z c= (dom cot) /\ (dom arctan) by A1, VALUED_1:def_4;
then A4: Z c= dom cot by XBOOLE_1:18;
 for x being Real st x in Z holds
cot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cot is_differentiable_in x )
assume  x in Z ; ::_thesis: cot is_differentiable_in x
then  sin . x <> 0 by A4, FDIFF_8:2;
hence  cot is_differentiable_in x by FDIFF_7:47; ::_thesis: verum
end;
then A5: cot is_differentiable_on Z by A4, FDIFF_1:9;
 for x being Real st x in Z holds
((cot (#) arctan) `| Z) . x = (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((cot (#) arctan) `| Z) . x = (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) )
assume A6: x in Z ; ::_thesis: ((cot (#) arctan) `| Z) . x = (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2)))
then A7: sin . x <> 0 by A4, FDIFF_8:2;
((cot (#) arctan) `| Z) . x = ((arctan . x) * (diff (cot,x))) + ((cot . x) * (diff (arctan,x))) by A1, A5, A3, A6, FDIFF_1:21
.= ((arctan . x) * (- (1 / ((sin . x) ^2)))) + ((cot . x) * (diff (arctan,x))) by A7, FDIFF_7:47
.= (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) * ((arctan `| Z) . x)) by A3, A6, FDIFF_1:def_7
.= (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) * (1 / (1 + (x ^2)))) by A2, A6, SIN_COS9:81
.= (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) ;
hence  ((cot (#) arctan) `| Z) . x = (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( cot (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) arctan) `| Z) . x = (- ((arctan . x) / ((sin . x) ^2))) + ((cot . x) / (1 + (x ^2))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:32
for Z being open Subset of REAL st Z c= dom (cot (#) arccot) & Z c= ].(- 1),1.[ holds
( cot (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) arccot) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cot (#) arccot) & Z c= ].(- 1),1.[ implies ( cot (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) arccot) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (cot (#) arccot) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cot (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) arccot) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) ) )
A3: arccot is_differentiable_on Z by A2, SIN_COS9:82;
 Z c= (dom cot) /\ (dom arccot) by A1, VALUED_1:def_4;
then A4: Z c= dom cot by XBOOLE_1:18;
 for x being Real st x in Z holds
cot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cot is_differentiable_in x )
assume  x in Z ; ::_thesis: cot is_differentiable_in x
then  sin . x <> 0 by A4, FDIFF_8:2;
hence  cot is_differentiable_in x by FDIFF_7:47; ::_thesis: verum
end;
then A5: cot is_differentiable_on Z by A4, FDIFF_1:9;
 for x being Real st x in Z holds
((cot (#) arccot) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((cot (#) arccot) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) )
assume A6: x in Z ; ::_thesis: ((cot (#) arccot) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2)))
then A7: sin . x <> 0 by A4, FDIFF_8:2;
((cot (#) arccot) `| Z) . x = ((arccot . x) * (diff (cot,x))) + ((cot . x) * (diff (arccot,x))) by A1, A5, A3, A6, FDIFF_1:21
.= ((arccot . x) * (- (1 / ((sin . x) ^2)))) + ((cot . x) * (diff (arccot,x))) by A7, FDIFF_7:47
.= (- ((arccot . x) / ((sin . x) ^2))) + ((cot . x) * ((arccot `| Z) . x)) by A3, A6, FDIFF_1:def_7
.= (- ((arccot . x) / ((sin . x) ^2))) + ((cot . x) * (- (1 / (1 + (x ^2))))) by A2, A6, SIN_COS9:82
.= (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) ;
hence  ((cot (#) arccot) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( cot (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) arccot) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2))) - ((cot . x) / (1 + (x ^2))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:33
for Z being open Subset of REAL st Z c= dom (sec (#) arctan) & Z c= ].(- 1),1.[ holds
( sec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) arctan) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sec (#) arctan) & Z c= ].(- 1),1.[ implies ( sec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) arctan) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) ) ) )
assume that
A1: Z c= dom (sec (#) arctan) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) arctan) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) ) )
A3: arctan is_differentiable_on Z by A2, SIN_COS9:81;
 Z c= (dom sec) /\ (dom arctan) by A1, VALUED_1:def_4;
then A4: Z c= dom sec by XBOOLE_1:18;
 for x being Real st x in Z holds
sec is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies sec is_differentiable_in x )
assume  x in Z ; ::_thesis: sec is_differentiable_in x
then  cos . x <> 0 by A4, RFUNCT_1:3;
hence  sec is_differentiable_in x by FDIFF_9:1; ::_thesis: verum
end;
then A5: sec is_differentiable_on Z by A4, FDIFF_1:9;
 for x being Real st x in Z holds
((sec (#) arctan) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies ((sec (#) arctan) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) )
assume A6: x in Z ; ::_thesis: ((sec (#) arctan) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2))))
then A7: cos . x <> 0 by A4, RFUNCT_1:3;
((sec (#) arctan) `| Z) . x = ((arctan . x) * (diff (sec,x))) + ((sec . x) * (diff (arctan,x))) by A1, A5, A3, A6, FDIFF_1:21
.= ((arctan . x) * ((sin . x) / ((cos . x) ^2))) + ((sec . x) * (diff (arctan,x))) by A7, FDIFF_9:1
.= (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + ((sec . x) * ((arctan `| Z) . x)) by A3, A6, FDIFF_1:def_7
.= (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + ((sec . x) * (1 / (1 + (x ^2)))) by A2, A6, SIN_COS9:81
.= (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + ((1 / (cos . x)) * (1 / (1 + (x ^2)))) by A4, A6, RFUNCT_1:def_2
.= (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) by XCMPLX_1:102 ;
hence  ((sec (#) arctan) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) ; ::_thesis: verum
end;
hence  ( sec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) arctan) `| Z) . x = (((sin . x) * (arctan . x)) / ((cos . x) ^2)) + (1 / ((cos . x) * (1 + (x ^2)))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:34
for Z being open Subset of REAL st Z c= dom (sec (#) arccot) & Z c= ].(- 1),1.[ holds
( sec (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) arccot) `| Z) . x = (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sec (#) arccot) & Z c= ].(- 1),1.[ implies ( sec (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) arccot) `| Z) . x = (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) ) ) )
assume that
A1: Z c= dom (sec (#) arccot) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sec (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) arccot) `| Z) . x = (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) ) )
A3: arccot is_differentiable_on Z by A2, SIN_COS9:82;
 Z c= (dom sec) /\ (dom arccot) by A1, VALUED_1:def_4;
then A4: Z c= dom sec by XBOOLE_1:18;
 for x being Real st x in Z holds
sec is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies sec is_differentiable_in x )
assume  x in Z ; ::_thesis: sec is_differentiable_in x
then  cos . x <> 0 by A4, RFUNCT_1:3;
hence  sec is_differentiable_in x by FDIFF_9:1; ::_thesis: verum
end;
then A5: sec is_differentiable_on Z by A4, FDIFF_1:9;
 for x being Real st x in Z holds
((sec (#) arccot) `| Z) . x = (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies ((sec (#) arccot) `| Z) . x = (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) )
assume A6: x in Z ; ::_thesis: ((sec (#) arccot) `| Z) . x = (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2))))
then A7: cos . x <> 0 by A4, RFUNCT_1:3;
((sec (#) arccot) `| Z) . x = ((arccot . x) * (diff (sec,x))) + ((sec . x) * (diff (arccot,x))) by A1, A5, A3, A6, FDIFF_1:21
.= ((arccot . x) * ((sin . x) / ((cos . x) ^2))) + ((sec . x) * (diff (arccot,x))) by A7, FDIFF_9:1
.= (((sin . x) * (arccot . x)) / ((cos . x) ^2)) + ((sec . x) * ((arccot `| Z) . x)) by A3, A6, FDIFF_1:def_7
.= (((sin . x) * (arccot . x)) / ((cos . x) ^2)) + ((sec . x) * (- (1 / (1 + (x ^2))))) by A2, A6, SIN_COS9:82
.= (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - ((sec . x) * (1 / (1 + (x ^2))))
.= (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - ((1 / (cos . x)) * (1 / (1 + (x ^2)))) by A4, A6, RFUNCT_1:def_2
.= (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) by XCMPLX_1:102 ;
hence  ((sec (#) arccot) `| Z) . x = (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) ; ::_thesis: verum
end;
hence  ( sec (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) arccot) `| Z) . x = (((sin . x) * (arccot . x)) / ((cos . x) ^2)) - (1 / ((cos . x) * (1 + (x ^2)))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:35
for Z being open Subset of REAL st Z c= dom (cosec (#) arctan) & Z c= ].(- 1),1.[ holds
( cosec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cosec (#) arctan) & Z c= ].(- 1),1.[ implies ( cosec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ) ) )
assume that
A1: Z c= dom (cosec (#) arctan) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cosec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ) )
A3: arctan is_differentiable_on Z by A2, SIN_COS9:81;
 Z c= (dom cosec) /\ (dom arctan) by A1, VALUED_1:def_4;
then A4: Z c= dom cosec by XBOOLE_1:18;
 for x being Real st x in Z holds
cosec is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cosec is_differentiable_in x )
assume  x in Z ; ::_thesis: cosec is_differentiable_in x
then  sin . x <> 0 by A4, RFUNCT_1:3;
hence  cosec is_differentiable_in x by FDIFF_9:2; ::_thesis: verum
end;
then A5: cosec is_differentiable_on Z by A4, FDIFF_1:9;
 for x being Real st x in Z holds
((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies ((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) )
assume A6: x in Z ; ::_thesis: ((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2))))
then A7: sin . x <> 0 by A4, RFUNCT_1:3;
((cosec (#) arctan) `| Z) . x = ((arctan . x) * (diff (cosec,x))) + ((cosec . x) * (diff (arctan,x))) by A1, A5, A3, A6, FDIFF_1:21
.= ((arctan . x) * (- ((cos . x) / ((sin . x) ^2)))) + ((cosec . x) * (diff (arctan,x))) by A7, FDIFF_9:2
.= (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + ((cosec . x) * ((arctan `| Z) . x)) by A3, A6, FDIFF_1:def_7
.= (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + ((cosec . x) * (1 / (1 + (x ^2)))) by A2, A6, SIN_COS9:81
.= (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + ((1 / (sin . x)) * (1 / (1 + (x ^2)))) by A4, A6, RFUNCT_1:def_2
.= (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) by XCMPLX_1:102 ;
hence  ((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ; ::_thesis: verum
end;
hence  ( cosec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:36
for Z being open Subset of REAL st Z c= dom (cosec (#) arccot) & Z c= ].(- 1),1.[ holds
( cosec (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) arccot) `| Z) . x = (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cosec (#) arccot) & Z c= ].(- 1),1.[ implies ( cosec (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) arccot) `| Z) . x = (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) ) ) )
assume that
A1: Z c= dom (cosec (#) arccot) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cosec (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) arccot) `| Z) . x = (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) ) )
A3: arccot is_differentiable_on Z by A2, SIN_COS9:82;
 Z c= (dom cosec) /\ (dom arccot) by A1, VALUED_1:def_4;
then A4: Z c= dom cosec by XBOOLE_1:18;
 for x being Real st x in Z holds
cosec is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cosec is_differentiable_in x )
assume  x in Z ; ::_thesis: cosec is_differentiable_in x
then  sin . x <> 0 by A4, RFUNCT_1:3;
hence  cosec is_differentiable_in x by FDIFF_9:2; ::_thesis: verum
end;
then A5: cosec is_differentiable_on Z by A4, FDIFF_1:9;
 for x being Real st x in Z holds
((cosec (#) arccot) `| Z) . x = (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies ((cosec (#) arccot) `| Z) . x = (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) )
assume A6: x in Z ; ::_thesis: ((cosec (#) arccot) `| Z) . x = (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2))))
then A7: sin . x <> 0 by A4, RFUNCT_1:3;
((cosec (#) arccot) `| Z) . x = ((arccot . x) * (diff (cosec,x))) + ((cosec . x) * (diff (arccot,x))) by A1, A5, A3, A6, FDIFF_1:21
.= ((arccot . x) * (- ((cos . x) / ((sin . x) ^2)))) + ((cosec . x) * (diff (arccot,x))) by A7, FDIFF_9:2
.= (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) + ((cosec . x) * ((arccot `| Z) . x)) by A3, A6, FDIFF_1:def_7
.= (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) + ((cosec . x) * (- (1 / (1 + (x ^2))))) by A2, A6, SIN_COS9:82
.= (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - ((cosec . x) * (1 / (1 + (x ^2))))
.= (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - ((1 / (sin . x)) * (1 / (1 + (x ^2)))) by A4, A6, RFUNCT_1:def_2
.= (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) by XCMPLX_1:102 ;
hence  ((cosec (#) arccot) `| Z) . x = (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) ; ::_thesis: verum
end;
hence  ( cosec (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) arccot) `| Z) . x = (- (((cos . x) * (arccot . x)) / ((sin . x) ^2))) - (1 / ((sin . x) * (1 + (x ^2)))) ) ) by A1, A5, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem Th37: :: FDIFF_11:37
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( arctan + arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan + arccot) `| Z) . x = 0 ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( arctan + arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan + arccot) `| Z) . x = 0 ) ) )
assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( arctan + arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan + arccot) `| Z) . x = 0 ) )
then A2: arctan is_differentiable_on Z by SIN_COS9:81;
A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A4: Z c= dom arccot by A1, XBOOLE_1:1;
A5: arccot is_differentiable_on Z by A1, SIN_COS9:82;
 ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A1, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19;
then A6: Z c= dom (arctan + arccot) by VALUED_1:def_1;
 for x being Real st x in Z holds
((arctan + arccot) `| Z) . x = 0
proof
let x be Real; ::_thesis: ( x in Z implies ((arctan + arccot) `| Z) . x = 0 )
assume A7: x in Z ; ::_thesis: ((arctan + arccot) `| Z) . x = 0
then ((arctan + arccot) `| Z) . x = (diff (arctan,x)) + (diff (arccot,x)) by A6, A2, A5, FDIFF_1:18
.= ((arctan `| Z) . x) + (diff (arccot,x)) by A2, A7, FDIFF_1:def_7
.= (1 / (1 + (x ^2))) + (diff (arccot,x)) by A1, A7, SIN_COS9:81
.= (1 / (1 + (x ^2))) + ((arccot `| Z) . x) by A5, A7, FDIFF_1:def_7
.= (1 / (1 + (x ^2))) + (- (1 / (1 + (x ^2)))) by A1, A7, SIN_COS9:82
.= 0 ;
hence  ((arctan + arccot) `| Z) . x = 0 ; ::_thesis: verum
end;
hence  ( arctan + arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan + arccot) `| Z) . x = 0 ) ) by A6, A2, A5, FDIFF_1:18; ::_thesis: verum
end;

theorem Th38: :: FDIFF_11:38
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( arctan - arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( arctan - arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) ) ) )
assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( arctan - arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) ) )
then A2: arctan is_differentiable_on Z by SIN_COS9:81;
A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A4: Z c= dom arccot by A1, XBOOLE_1:1;
A5: arccot is_differentiable_on Z by A1, SIN_COS9:82;
 ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A1, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19;
then A6: Z c= dom (arctan - arccot) by VALUED_1:12;
 for x being Real st x in Z holds
((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2))
proof
let x be Real; ::_thesis: ( x in Z implies ((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) )
assume A7: x in Z ; ::_thesis: ((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2))
then ((arctan - arccot) `| Z) . x = (diff (arctan,x)) - (diff (arccot,x)) by A6, A2, A5, FDIFF_1:19
.= ((arctan `| Z) . x) - (diff (arccot,x)) by A2, A7, FDIFF_1:def_7
.= (1 / (1 + (x ^2))) - (diff (arccot,x)) by A1, A7, SIN_COS9:81
.= (1 / (1 + (x ^2))) - ((arccot `| Z) . x) by A5, A7, FDIFF_1:def_7
.= (1 / (1 + (x ^2))) - (- (1 / (1 + (x ^2)))) by A1, A7, SIN_COS9:82
.= 2 / (1 + (x ^2)) ;
hence  ((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) ; ::_thesis: verum
end;
hence  ( arctan - arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) ) ) by A6, A2, A5, FDIFF_1:19; ::_thesis: verum
end;

theorem :: FDIFF_11:39
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( sin (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( sin (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x)) ) ) )
 for x being Real st x in Z holds
sin is_differentiable_in x by SIN_COS:64;
then A1: sin is_differentiable_on Z by FDIFF_1:9, SIN_COS:24;
assume A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sin (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x)) ) )
then A3: arctan + arccot is_differentiable_on Z by Th37;
A4: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A5: Z c= dom arccot by A2, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A4, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A2, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A5, XBOOLE_1:19;
then A6: Z c= dom (arctan + arccot) by VALUED_1:def_1;
then  Z c= (dom sin) /\ (dom (arctan + arccot)) by SIN_COS:24, XBOOLE_1:19;
then A7: Z c= dom (sin (#) (arctan + arccot)) by VALUED_1:def_4;
 for x being Real st x in Z holds
((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x))
proof
let x be Real; ::_thesis: ( x in Z implies ((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x)) )
assume A8: x in Z ; ::_thesis: ((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x))
then ((sin (#) (arctan + arccot)) `| Z) . x = (((arctan + arccot) . x) * (diff (sin,x))) + ((sin . x) * (diff ((arctan + arccot),x))) by A7, A1, A3, FDIFF_1:21
.= (((arctan . x) + (arccot . x)) * (diff (sin,x))) + ((sin . x) * (diff ((arctan + arccot),x))) by A6, A8, VALUED_1:def_1
.= (((arctan . x) + (arccot . x)) * (cos . x)) + ((sin . x) * (diff ((arctan + arccot),x))) by SIN_COS:64
.= (((arctan . x) + (arccot . x)) * (cos . x)) + ((sin . x) * (((arctan + arccot) `| Z) . x)) by A3, A8, FDIFF_1:def_7
.= (((arctan . x) + (arccot . x)) * (cos . x)) + ((sin . x) * 0) by A2, A8, Th37
.= (cos . x) * ((arctan . x) + (arccot . x)) ;
hence  ((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x)) ; ::_thesis: verum
end;
hence  ( sin (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x)) ) ) by A7, A1, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:40
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( sin (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) (arctan - arccot)) `| Z) . x = ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( sin (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) (arctan - arccot)) `| Z) . x = ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) ) ) )
 for x being Real st x in Z holds
sin is_differentiable_in x by SIN_COS:64;
then A1: sin is_differentiable_on Z by FDIFF_1:9, SIN_COS:24;
assume A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sin (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) (arctan - arccot)) `| Z) . x = ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) ) )
then A3: arctan - arccot is_differentiable_on Z by Th38;
A4: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A5: Z c= dom arccot by A2, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A4, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A2, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A5, XBOOLE_1:19;
then A6: Z c= dom (arctan - arccot) by VALUED_1:12;
then  Z c= (dom sin) /\ (dom (arctan - arccot)) by SIN_COS:24, XBOOLE_1:19;
then A7: Z c= dom (sin (#) (arctan - arccot)) by VALUED_1:def_4;
 for x being Real st x in Z holds
((sin (#) (arctan - arccot)) `| Z) . x = ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((sin (#) (arctan - arccot)) `| Z) . x = ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) )
assume A8: x in Z ; ::_thesis: ((sin (#) (arctan - arccot)) `| Z) . x = ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2)))
then ((sin (#) (arctan - arccot)) `| Z) . x = (((arctan - arccot) . x) * (diff (sin,x))) + ((sin . x) * (diff ((arctan - arccot),x))) by A7, A1, A3, FDIFF_1:21
.= (((arctan . x) - (arccot . x)) * (diff (sin,x))) + ((sin . x) * (diff ((arctan - arccot),x))) by A6, A8, VALUED_1:13
.= (((arctan . x) - (arccot . x)) * (cos . x)) + ((sin . x) * (diff ((arctan - arccot),x))) by SIN_COS:64
.= (((arctan . x) - (arccot . x)) * (cos . x)) + ((sin . x) * (((arctan - arccot) `| Z) . x)) by A3, A8, FDIFF_1:def_7
.= (((arctan . x) - (arccot . x)) * (cos . x)) + ((sin . x) * (2 / (1 + (x ^2)))) by A2, A8, Th38
.= ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) ;
hence  ((sin (#) (arctan - arccot)) `| Z) . x = ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( sin (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) (arctan - arccot)) `| Z) . x = ((cos . x) * ((arctan . x) - (arccot . x))) + ((2 * (sin . x)) / (1 + (x ^2))) ) ) by A7, A1, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:41
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( cos (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) (arctan + arccot)) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( cos (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) (arctan + arccot)) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x))) ) ) )
 for x being Real st x in Z holds
cos is_differentiable_in x by SIN_COS:63;
then A1: cos is_differentiable_on Z by FDIFF_1:9, SIN_COS:24;
assume A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cos (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) (arctan + arccot)) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x))) ) )
then A3: arctan + arccot is_differentiable_on Z by Th37;
A4: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A5: Z c= dom arccot by A2, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A4, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A2, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A5, XBOOLE_1:19;
then A6: Z c= dom (arctan + arccot) by VALUED_1:def_1;
then  Z c= (dom cos) /\ (dom (arctan + arccot)) by SIN_COS:24, XBOOLE_1:19;
then A7: Z c= dom (cos (#) (arctan + arccot)) by VALUED_1:def_4;
 for x being Real st x in Z holds
((cos (#) (arctan + arccot)) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x)))
proof
let x be Real; ::_thesis: ( x in Z implies ((cos (#) (arctan + arccot)) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x))) )
assume A8: x in Z ; ::_thesis: ((cos (#) (arctan + arccot)) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x)))
then ((cos (#) (arctan + arccot)) `| Z) . x = (((arctan + arccot) . x) * (diff (cos,x))) + ((cos . x) * (diff ((arctan + arccot),x))) by A7, A1, A3, FDIFF_1:21
.= (((arctan . x) + (arccot . x)) * (diff (cos,x))) + ((cos . x) * (diff ((arctan + arccot),x))) by A6, A8, VALUED_1:def_1
.= (((arctan . x) + (arccot . x)) * (- (sin . x))) + ((cos . x) * (diff ((arctan + arccot),x))) by SIN_COS:63
.= (((arctan . x) + (arccot . x)) * (- (sin . x))) + ((cos . x) * (((arctan + arccot) `| Z) . x)) by A3, A8, FDIFF_1:def_7
.= (- (((arctan . x) + (arccot . x)) * (sin . x))) + ((cos . x) * 0) by A2, A8, Th37
.= - ((sin . x) * ((arctan . x) + (arccot . x))) ;
hence  ((cos (#) (arctan + arccot)) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x))) ; ::_thesis: verum
end;
hence  ( cos (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) (arctan + arccot)) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x))) ) ) by A7, A1, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:42
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( cos (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) (arctan - arccot)) `| Z) . x = (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( cos (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) (arctan - arccot)) `| Z) . x = (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) ) ) )
 for x being Real st x in Z holds
cos is_differentiable_in x by SIN_COS:63;
then A1: cos is_differentiable_on Z by FDIFF_1:9, SIN_COS:24;
assume A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cos (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) (arctan - arccot)) `| Z) . x = (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) ) )
then A3: arctan - arccot is_differentiable_on Z by Th38;
A4: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A5: Z c= dom arccot by A2, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A4, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A2, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A5, XBOOLE_1:19;
then A6: Z c= dom (arctan - arccot) by VALUED_1:12;
then  Z c= (dom cos) /\ (dom (arctan - arccot)) by SIN_COS:24, XBOOLE_1:19;
then A7: Z c= dom (cos (#) (arctan - arccot)) by VALUED_1:def_4;
 for x being Real st x in Z holds
((cos (#) (arctan - arccot)) `| Z) . x = (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((cos (#) (arctan - arccot)) `| Z) . x = (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) )
assume A8: x in Z ; ::_thesis: ((cos (#) (arctan - arccot)) `| Z) . x = (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2)))
then ((cos (#) (arctan - arccot)) `| Z) . x = (((arctan - arccot) . x) * (diff (cos,x))) + ((cos . x) * (diff ((arctan - arccot),x))) by A7, A1, A3, FDIFF_1:21
.= (((arctan . x) - (arccot . x)) * (diff (cos,x))) + ((cos . x) * (diff ((arctan - arccot),x))) by A6, A8, VALUED_1:13
.= (((arctan . x) - (arccot . x)) * (- (sin . x))) + ((cos . x) * (diff ((arctan - arccot),x))) by SIN_COS:63
.= (((arctan . x) - (arccot . x)) * (- (sin . x))) + ((cos . x) * (((arctan - arccot) `| Z) . x)) by A3, A8, FDIFF_1:def_7
.= (- (((arctan . x) - (arccot . x)) * (sin . x))) + ((cos . x) * (2 / (1 + (x ^2)))) by A2, A8, Th38
.= (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) ;
hence  ((cos (#) (arctan - arccot)) `| Z) . x = (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( cos (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) (arctan - arccot)) `| Z) . x = (- ((sin . x) * ((arctan . x) - (arccot . x)))) + ((2 * (cos . x)) / (1 + (x ^2))) ) ) by A7, A1, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:43
for Z being open Subset of REAL st Z c= dom tan & Z c= ].(- 1),1.[ holds
( tan (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) (arctan + arccot)) `| Z) . x = ((arctan . x) + (arccot . x)) / ((cos . x) ^2) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom tan & Z c= ].(- 1),1.[ implies ( tan (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) (arctan + arccot)) `| Z) . x = ((arctan . x) + (arccot . x)) / ((cos . x) ^2) ) ) )
assume that
A1: Z c= dom tan and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( tan (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) (arctan + arccot)) `| Z) . x = ((arctan . x) + (arccot . x)) / ((cos . x) ^2) ) )
A3: arctan + arccot is_differentiable_on Z by A2, Th37;
 for x being Real st x in Z holds
tan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies tan is_differentiable_in x )
assume  x in Z ; ::_thesis: tan is_differentiable_in x
then  cos . x <> 0 by A1, FDIFF_8:1;
hence  tan is_differentiable_in x by FDIFF_7:46; ::_thesis: verum
end;
then A4: tan is_differentiable_on Z by A1, FDIFF_1:9;
A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A6: Z c= dom arccot by A2, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A2, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19;
then A7: Z c= dom (arctan + arccot) by VALUED_1:def_1;
then  Z c= (dom tan) /\ (dom (arctan + arccot)) by A1, XBOOLE_1:19;
then A8: Z c= dom (tan (#) (arctan + arccot)) by VALUED_1:def_4;
 for x being Real st x in Z holds
((tan (#) (arctan + arccot)) `| Z) . x = ((arctan . x) + (arccot . x)) / ((cos . x) ^2)
proof
let x be Real; ::_thesis: ( x in Z implies ((tan (#) (arctan + arccot)) `| Z) . x = ((arctan . x) + (arccot . x)) / ((cos . x) ^2) )
assume A9: x in Z ; ::_thesis: ((tan (#) (arctan + arccot)) `| Z) . x = ((arctan . x) + (arccot . x)) / ((cos . x) ^2)
then A10: cos . x <> 0 by A1, FDIFF_8:1;
((tan (#) (arctan + arccot)) `| Z) . x = (((arctan + arccot) . x) * (diff (tan,x))) + ((tan . x) * (diff ((arctan + arccot),x))) by A8, A4, A3, A9, FDIFF_1:21
.= (((arctan . x) + (arccot . x)) * (diff (tan,x))) + ((tan . x) * (diff ((arctan + arccot),x))) by A7, A9, VALUED_1:def_1
.= (((arctan . x) + (arccot . x)) * (1 / ((cos . x) ^2))) + ((tan . x) * (diff ((arctan + arccot),x))) by A10, FDIFF_7:46
.= (((arctan . x) + (arccot . x)) / ((cos . x) ^2)) + ((tan . x) * (((arctan + arccot) `| Z) . x)) by A3, A9, FDIFF_1:def_7
.= (((arctan . x) + (arccot . x)) / ((cos . x) ^2)) + ((tan . x) * 0) by A2, A9, Th37
.= ((arctan . x) + (arccot . x)) / ((cos . x) ^2) ;
hence  ((tan (#) (arctan + arccot)) `| Z) . x = ((arctan . x) + (arccot . x)) / ((cos . x) ^2) ; ::_thesis: verum
end;
hence  ( tan (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) (arctan + arccot)) `| Z) . x = ((arctan . x) + (arccot . x)) / ((cos . x) ^2) ) ) by A8, A4, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:44
for Z being open Subset of REAL st Z c= dom tan & Z c= ].(- 1),1.[ holds
( tan (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) (arctan - arccot)) `| Z) . x = (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom tan & Z c= ].(- 1),1.[ implies ( tan (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) (arctan - arccot)) `| Z) . x = (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom tan and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( tan (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) (arctan - arccot)) `| Z) . x = (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) ) )
A3: arctan - arccot is_differentiable_on Z by A2, Th38;
 for x being Real st x in Z holds
tan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies tan is_differentiable_in x )
assume  x in Z ; ::_thesis: tan is_differentiable_in x
then  cos . x <> 0 by A1, FDIFF_8:1;
hence  tan is_differentiable_in x by FDIFF_7:46; ::_thesis: verum
end;
then A4: tan is_differentiable_on Z by A1, FDIFF_1:9;
A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A6: Z c= dom arccot by A2, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A2, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19;
then A7: Z c= dom (arctan - arccot) by VALUED_1:12;
then  Z c= (dom tan) /\ (dom (arctan - arccot)) by A1, XBOOLE_1:19;
then A8: Z c= dom (tan (#) (arctan - arccot)) by VALUED_1:def_4;
 for x being Real st x in Z holds
((tan (#) (arctan - arccot)) `| Z) . x = (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((tan (#) (arctan - arccot)) `| Z) . x = (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) )
assume A9: x in Z ; ::_thesis: ((tan (#) (arctan - arccot)) `| Z) . x = (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2)))
then A10: cos . x <> 0 by A1, FDIFF_8:1;
((tan (#) (arctan - arccot)) `| Z) . x = (((arctan - arccot) . x) * (diff (tan,x))) + ((tan . x) * (diff ((arctan - arccot),x))) by A8, A4, A3, A9, FDIFF_1:21
.= (((arctan . x) - (arccot . x)) * (diff (tan,x))) + ((tan . x) * (diff ((arctan - arccot),x))) by A7, A9, VALUED_1:13
.= (((arctan . x) - (arccot . x)) * (1 / ((cos . x) ^2))) + ((tan . x) * (diff ((arctan - arccot),x))) by A10, FDIFF_7:46
.= (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((tan . x) * (((arctan - arccot) `| Z) . x)) by A3, A9, FDIFF_1:def_7
.= (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((tan . x) * (2 / (1 + (x ^2)))) by A2, A9, Th38
.= (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) ;
hence  ((tan (#) (arctan - arccot)) `| Z) . x = (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( tan (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) (arctan - arccot)) `| Z) . x = (((arctan . x) - (arccot . x)) / ((cos . x) ^2)) + ((2 * (tan . x)) / (1 + (x ^2))) ) ) by A8, A4, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:45
for Z being open Subset of REAL st Z c= dom cot & Z c= ].(- 1),1.[ holds
( cot (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) (arctan + arccot)) `| Z) . x = - (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom cot & Z c= ].(- 1),1.[ implies ( cot (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) (arctan + arccot)) `| Z) . x = - (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) ) ) )
assume that
A1: Z c= dom cot and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cot (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) (arctan + arccot)) `| Z) . x = - (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) ) )
A3: arctan + arccot is_differentiable_on Z by A2, Th37;
 for x being Real st x in Z holds
cot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cot is_differentiable_in x )
assume  x in Z ; ::_thesis: cot is_differentiable_in x
then  sin . x <> 0 by A1, FDIFF_8:2;
hence  cot is_differentiable_in x by FDIFF_7:47; ::_thesis: verum
end;
then A4: cot is_differentiable_on Z by A1, FDIFF_1:9;
A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A6: Z c= dom arccot by A2, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A2, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19;
then A7: Z c= dom (arctan + arccot) by VALUED_1:def_1;
then  Z c= (dom cot) /\ (dom (arctan + arccot)) by A1, XBOOLE_1:19;
then A8: Z c= dom (cot (#) (arctan + arccot)) by VALUED_1:def_4;
 for x being Real st x in Z holds
((cot (#) (arctan + arccot)) `| Z) . x = - (((arctan . x) + (arccot . x)) / ((sin . x) ^2))
proof
let x be Real; ::_thesis: ( x in Z implies ((cot (#) (arctan + arccot)) `| Z) . x = - (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) )
assume A9: x in Z ; ::_thesis: ((cot (#) (arctan + arccot)) `| Z) . x = - (((arctan . x) + (arccot . x)) / ((sin . x) ^2))
then A10: sin . x <> 0 by A1, FDIFF_8:2;
((cot (#) (arctan + arccot)) `| Z) . x = (((arctan + arccot) . x) * (diff (cot,x))) + ((cot . x) * (diff ((arctan + arccot),x))) by A8, A4, A3, A9, FDIFF_1:21
.= (((arctan . x) + (arccot . x)) * (diff (cot,x))) + ((cot . x) * (diff ((arctan + arccot),x))) by A7, A9, VALUED_1:def_1
.= (((arctan . x) + (arccot . x)) * (- (1 / ((sin . x) ^2)))) + ((cot . x) * (diff ((arctan + arccot),x))) by A10, FDIFF_7:47
.= (- (((arctan . x) + (arccot . x)) / ((sin . x) ^2))) + ((cot . x) * (((arctan + arccot) `| Z) . x)) by A3, A9, FDIFF_1:def_7
.= (- (((arctan . x) + (arccot . x)) / ((sin . x) ^2))) + ((cot . x) * 0) by A2, A9, Th37
.= - (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) ;
hence  ((cot (#) (arctan + arccot)) `| Z) . x = - (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) ; ::_thesis: verum
end;
hence  ( cot (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) (arctan + arccot)) `| Z) . x = - (((arctan . x) + (arccot . x)) / ((sin . x) ^2)) ) ) by A8, A4, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:46
for Z being open Subset of REAL st Z c= dom cot & Z c= ].(- 1),1.[ holds
( cot (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom cot & Z c= ].(- 1),1.[ implies ( cot (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom cot and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cot (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ) )
A3: arctan - arccot is_differentiable_on Z by A2, Th38;
 for x being Real st x in Z holds
cot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cot is_differentiable_in x )
assume  x in Z ; ::_thesis: cot is_differentiable_in x
then  sin . x <> 0 by A1, FDIFF_8:2;
hence  cot is_differentiable_in x by FDIFF_7:47; ::_thesis: verum
end;
then A4: cot is_differentiable_on Z by A1, FDIFF_1:9;
A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A6: Z c= dom arccot by A2, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A2, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19;
then A7: Z c= dom (arctan - arccot) by VALUED_1:12;
then  Z c= (dom cot) /\ (dom (arctan - arccot)) by A1, XBOOLE_1:19;
then A8: Z c= dom (cot (#) (arctan - arccot)) by VALUED_1:def_4;
 for x being Real st x in Z holds
((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) )
assume A9: x in Z ; ::_thesis: ((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2)))
then A10: sin . x <> 0 by A1, FDIFF_8:2;
((cot (#) (arctan - arccot)) `| Z) . x = (((arctan - arccot) . x) * (diff (cot,x))) + ((cot . x) * (diff ((arctan - arccot),x))) by A8, A4, A3, A9, FDIFF_1:21
.= (((arctan . x) - (arccot . x)) * (diff (cot,x))) + ((cot . x) * (diff ((arctan - arccot),x))) by A7, A9, VALUED_1:13
.= (((arctan . x) - (arccot . x)) * (- (1 / ((sin . x) ^2)))) + ((cot . x) * (diff ((arctan - arccot),x))) by A10, FDIFF_7:47
.= (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((cot . x) * (((arctan - arccot) `| Z) . x)) by A3, A9, FDIFF_1:def_7
.= (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((cot . x) * (2 / (1 + (x ^2)))) by A2, A9, Th38
.= (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ;
hence  ((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( cot (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ) ) by A8, A4, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:47
for Z being open Subset of REAL st Z c= dom sec & Z c= ].(- 1),1.[ holds
( sec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) (arctan + arccot)) `| Z) . x = (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom sec & Z c= ].(- 1),1.[ implies ( sec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) (arctan + arccot)) `| Z) . x = (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) ) ) )
assume that
A1: Z c= dom sec and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) (arctan + arccot)) `| Z) . x = (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) ) )
A3: arctan + arccot is_differentiable_on Z by A2, Th37;
 for x being Real st x in Z holds
sec is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies sec is_differentiable_in x )
assume  x in Z ; ::_thesis: sec is_differentiable_in x
then  cos . x <> 0 by A1, RFUNCT_1:3;
hence  sec is_differentiable_in x by FDIFF_9:1; ::_thesis: verum
end;
then A4: sec is_differentiable_on Z by A1, FDIFF_1:9;
A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A6: Z c= dom arccot by A2, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A2, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19;
then A7: Z c= dom (arctan + arccot) by VALUED_1:def_1;
then  Z c= (dom sec) /\ (dom (arctan + arccot)) by A1, XBOOLE_1:19;
then A8: Z c= dom (sec (#) (arctan + arccot)) by VALUED_1:def_4;
 for x being Real st x in Z holds
((sec (#) (arctan + arccot)) `| Z) . x = (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2)
proof
let x be Real; ::_thesis: ( x in Z implies ((sec (#) (arctan + arccot)) `| Z) . x = (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) )
assume A9: x in Z ; ::_thesis: ((sec (#) (arctan + arccot)) `| Z) . x = (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2)
then A10: cos . x <> 0 by A1, RFUNCT_1:3;
((sec (#) (arctan + arccot)) `| Z) . x = (((arctan + arccot) . x) * (diff (sec,x))) + ((sec . x) * (diff ((arctan + arccot),x))) by A8, A4, A3, A9, FDIFF_1:21
.= (((arctan . x) + (arccot . x)) * (diff (sec,x))) + ((sec . x) * (diff ((arctan + arccot),x))) by A7, A9, VALUED_1:def_1
.= (((arctan . x) + (arccot . x)) * ((sin . x) / ((cos . x) ^2))) + ((sec . x) * (diff ((arctan + arccot),x))) by A10, FDIFF_9:1
.= ((((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((sec . x) * (((arctan + arccot) `| Z) . x)) by A3, A9, FDIFF_1:def_7
.= ((((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((sec . x) * 0) by A2, A9, Th37
.= (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) ;
hence  ((sec (#) (arctan + arccot)) `| Z) . x = (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) ; ::_thesis: verum
end;
hence  ( sec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) (arctan + arccot)) `| Z) . x = (((arctan . x) + (arccot . x)) * (sin . x)) / ((cos . x) ^2) ) ) by A8, A4, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:48
for Z being open Subset of REAL st Z c= dom sec & Z c= ].(- 1),1.[ holds
( sec (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) (arctan - arccot)) `| Z) . x = ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom sec & Z c= ].(- 1),1.[ implies ( sec (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) (arctan - arccot)) `| Z) . x = ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom sec and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sec (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) (arctan - arccot)) `| Z) . x = ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) ) )
A3: arctan - arccot is_differentiable_on Z by A2, Th38;
 for x being Real st x in Z holds
sec is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies sec is_differentiable_in x )
assume  x in Z ; ::_thesis: sec is_differentiable_in x
then  cos . x <> 0 by A1, RFUNCT_1:3;
hence  sec is_differentiable_in x by FDIFF_9:1; ::_thesis: verum
end;
then A4: sec is_differentiable_on Z by A1, FDIFF_1:9;
A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A6: Z c= dom arccot by A2, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A2, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19;
then A7: Z c= dom (arctan - arccot) by VALUED_1:12;
then  Z c= (dom sec) /\ (dom (arctan - arccot)) by A1, XBOOLE_1:19;
then A8: Z c= dom (sec (#) (arctan - arccot)) by VALUED_1:def_4;
 for x being Real st x in Z holds
((sec (#) (arctan - arccot)) `| Z) . x = ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((sec (#) (arctan - arccot)) `| Z) . x = ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) )
assume A9: x in Z ; ::_thesis: ((sec (#) (arctan - arccot)) `| Z) . x = ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2)))
then A10: cos . x <> 0 by A1, RFUNCT_1:3;
((sec (#) (arctan - arccot)) `| Z) . x = (((arctan - arccot) . x) * (diff (sec,x))) + ((sec . x) * (diff ((arctan - arccot),x))) by A8, A4, A3, A9, FDIFF_1:21
.= (((arctan . x) - (arccot . x)) * (diff (sec,x))) + ((sec . x) * (diff ((arctan - arccot),x))) by A7, A9, VALUED_1:13
.= (((arctan . x) - (arccot . x)) * ((sin . x) / ((cos . x) ^2))) + ((sec . x) * (diff ((arctan - arccot),x))) by A10, FDIFF_9:1
.= ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((sec . x) * (((arctan - arccot) `| Z) . x)) by A3, A9, FDIFF_1:def_7
.= ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((sec . x) * (2 / (1 + (x ^2)))) by A2, A9, Th38
.= ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) ;
hence  ((sec (#) (arctan - arccot)) `| Z) . x = ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( sec (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sec (#) (arctan - arccot)) `| Z) . x = ((((arctan . x) - (arccot . x)) * (sin . x)) / ((cos . x) ^2)) + ((2 * (sec . x)) / (1 + (x ^2))) ) ) by A8, A4, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:49
for Z being open Subset of REAL st Z c= dom cosec & Z c= ].(- 1),1.[ holds
( cosec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom cosec & Z c= ].(- 1),1.[ implies ( cosec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ) ) )
assume that
A1: Z c= dom cosec and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cosec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ) )
A3: arctan + arccot is_differentiable_on Z by A2, Th37;
 for x being Real st x in Z holds
cosec is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cosec is_differentiable_in x )
assume  x in Z ; ::_thesis: cosec is_differentiable_in x
then  sin . x <> 0 by A1, RFUNCT_1:3;
hence  cosec is_differentiable_in x by FDIFF_9:2; ::_thesis: verum
end;
then A4: cosec is_differentiable_on Z by A1, FDIFF_1:9;
A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A6: Z c= dom arccot by A2, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A2, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19;
then A7: Z c= dom (arctan + arccot) by VALUED_1:def_1;
then  Z c= (dom cosec) /\ (dom (arctan + arccot)) by A1, XBOOLE_1:19;
then A8: Z c= dom (cosec (#) (arctan + arccot)) by VALUED_1:def_4;
 for x being Real st x in Z holds
((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2))
proof
let x be Real; ::_thesis: ( x in Z implies ((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) )
assume A9: x in Z ; ::_thesis: ((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2))
then A10: sin . x <> 0 by A1, RFUNCT_1:3;
((cosec (#) (arctan + arccot)) `| Z) . x = (((arctan + arccot) . x) * (diff (cosec,x))) + ((cosec . x) * (diff ((arctan + arccot),x))) by A8, A4, A3, A9, FDIFF_1:21
.= (((arctan . x) + (arccot . x)) * (diff (cosec,x))) + ((cosec . x) * (diff ((arctan + arccot),x))) by A7, A9, VALUED_1:def_1
.= (((arctan . x) + (arccot . x)) * (- ((cos . x) / ((sin . x) ^2)))) + ((cosec . x) * (diff ((arctan + arccot),x))) by A10, FDIFF_9:2
.= (- ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((cosec . x) * (((arctan + arccot) `| Z) . x)) by A3, A9, FDIFF_1:def_7
.= (- ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((cosec . x) * 0) by A2, A9, Th37
.= - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ;
hence  ((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ; ::_thesis: verum
end;
hence  ( cosec (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) (arctan + arccot)) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2)) ) ) by A8, A4, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:50
for Z being open Subset of REAL st Z c= dom cosec & Z c= ].(- 1),1.[ holds
( cosec (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) (arctan - arccot)) `| Z) . x = (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom cosec & Z c= ].(- 1),1.[ implies ( cosec (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) (arctan - arccot)) `| Z) . x = (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom cosec and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cosec (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) (arctan - arccot)) `| Z) . x = (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) ) )
A3: arctan - arccot is_differentiable_on Z by A2, Th38;
 for x being Real st x in Z holds
cosec is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cosec is_differentiable_in x )
assume  x in Z ; ::_thesis: cosec is_differentiable_in x
then  sin . x <> 0 by A1, RFUNCT_1:3;
hence  cosec is_differentiable_in x by FDIFF_9:2; ::_thesis: verum
end;
then A4: cosec is_differentiable_on Z by A1, FDIFF_1:9;
A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A6: Z c= dom arccot by A2, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A2, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19;
then A7: Z c= dom (arctan - arccot) by VALUED_1:12;
then  Z c= (dom cosec) /\ (dom (arctan - arccot)) by A1, XBOOLE_1:19;
then A8: Z c= dom (cosec (#) (arctan - arccot)) by VALUED_1:def_4;
 for x being Real st x in Z holds
((cosec (#) (arctan - arccot)) `| Z) . x = (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((cosec (#) (arctan - arccot)) `| Z) . x = (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) )
assume A9: x in Z ; ::_thesis: ((cosec (#) (arctan - arccot)) `| Z) . x = (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2)))
then A10: sin . x <> 0 by A1, RFUNCT_1:3;
((cosec (#) (arctan - arccot)) `| Z) . x = (((arctan - arccot) . x) * (diff (cosec,x))) + ((cosec . x) * (diff ((arctan - arccot),x))) by A8, A4, A3, A9, FDIFF_1:21
.= (((arctan . x) - (arccot . x)) * (diff (cosec,x))) + ((cosec . x) * (diff ((arctan - arccot),x))) by A7, A9, VALUED_1:13
.= (((arctan . x) - (arccot . x)) * (- ((cos . x) / ((sin . x) ^2)))) + ((cosec . x) * (diff ((arctan - arccot),x))) by A10, FDIFF_9:2
.= (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((cosec . x) * (((arctan - arccot) `| Z) . x)) by A3, A9, FDIFF_1:def_7
.= (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((cosec . x) * (2 / (1 + (x ^2)))) by A2, A9, Th38
.= (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) ;
hence  ((cosec (#) (arctan - arccot)) `| Z) . x = (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( cosec (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) (arctan - arccot)) `| Z) . x = (- ((((arctan . x) - (arccot . x)) * (cos . x)) / ((sin . x) ^2))) + ((2 * (cosec . x)) / (1 + (x ^2))) ) ) by A8, A4, A3, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:51
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( exp_R (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) (arctan + arccot)) `| Z) . x = (exp_R . x) * ((arctan . x) + (arccot . x)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( exp_R (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) (arctan + arccot)) `| Z) . x = (exp_R . x) * ((arctan . x) + (arccot . x)) ) ) )
assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( exp_R (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) (arctan + arccot)) `| Z) . x = (exp_R . x) * ((arctan . x) + (arccot . x)) ) )
then A2: arctan + arccot is_differentiable_on Z by Th37;
A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A4: Z c= dom arccot by A1, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A1, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19;
then A5: Z c= dom (arctan + arccot) by VALUED_1:def_1;
then  Z c= (dom exp_R) /\ (dom (arctan + arccot)) by TAYLOR_1:16, XBOOLE_1:19;
then A6: Z c= dom (exp_R (#) (arctan + arccot)) by VALUED_1:def_4;
A7: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16;
 for x being Real st x in Z holds
((exp_R (#) (arctan + arccot)) `| Z) . x = (exp_R . x) * ((arctan . x) + (arccot . x))
proof
let x be Real; ::_thesis: ( x in Z implies ((exp_R (#) (arctan + arccot)) `| Z) . x = (exp_R . x) * ((arctan . x) + (arccot . x)) )
assume A8: x in Z ; ::_thesis: ((exp_R (#) (arctan + arccot)) `| Z) . x = (exp_R . x) * ((arctan . x) + (arccot . x))
then ((exp_R (#) (arctan + arccot)) `| Z) . x = (((arctan + arccot) . x) * (diff (exp_R,x))) + ((exp_R . x) * (diff ((arctan + arccot),x))) by A6, A7, A2, FDIFF_1:21
.= (((arctan . x) + (arccot . x)) * (diff (exp_R,x))) + ((exp_R . x) * (diff ((arctan + arccot),x))) by A5, A8, VALUED_1:def_1
.= (((arctan . x) + (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (diff ((arctan + arccot),x))) by TAYLOR_1:16
.= (((arctan . x) + (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (((arctan + arccot) `| Z) . x)) by A2, A8, FDIFF_1:def_7
.= (((arctan . x) + (arccot . x)) * (exp_R . x)) + ((exp_R . x) * 0) by A1, A8, Th37
.= (exp_R . x) * ((arctan . x) + (arccot . x)) ;
hence  ((exp_R (#) (arctan + arccot)) `| Z) . x = (exp_R . x) * ((arctan . x) + (arccot . x)) ; ::_thesis: verum
end;
hence  ( exp_R (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) (arctan + arccot)) `| Z) . x = (exp_R . x) * ((arctan . x) + (arccot . x)) ) ) by A6, A7, A2, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:52
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( exp_R (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( exp_R (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ) ) )
assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( exp_R (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ) )
then A2: arctan - arccot is_differentiable_on Z by Th38;
A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A4: Z c= dom arccot by A1, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A1, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19;
then A5: Z c= dom (arctan - arccot) by VALUED_1:12;
then  Z c= (dom exp_R) /\ (dom (arctan - arccot)) by TAYLOR_1:16, XBOOLE_1:19;
then A6: Z c= dom (exp_R (#) (arctan - arccot)) by VALUED_1:def_4;
A7: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16;
 for x being Real st x in Z holds
((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) )
assume A8: x in Z ; ::_thesis: ((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2)))
then ((exp_R (#) (arctan - arccot)) `| Z) . x = (((arctan - arccot) . x) * (diff (exp_R,x))) + ((exp_R . x) * (diff ((arctan - arccot),x))) by A6, A7, A2, FDIFF_1:21
.= (((arctan . x) - (arccot . x)) * (diff (exp_R,x))) + ((exp_R . x) * (diff ((arctan - arccot),x))) by A5, A8, VALUED_1:13
.= (((arctan . x) - (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (diff ((arctan - arccot),x))) by TAYLOR_1:16
.= (((arctan . x) - (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (((arctan - arccot) `| Z) . x)) by A2, A8, FDIFF_1:def_7
.= (((arctan . x) - (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (2 / (1 + (x ^2)))) by A1, A8, Th38
.= ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ;
hence  ((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( exp_R (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ) ) by A6, A7, A2, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:53
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( (arctan + arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( (arctan + arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ) ) )
assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( (arctan + arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ) )
then A2: arctan + arccot is_differentiable_on Z by Th37;
A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A4: Z c= dom arccot by A1, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A1, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19;
then A5: Z c= dom (arctan + arccot) by VALUED_1:def_1;
A6: ( exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
exp_R . x <> 0 ) ) by FDIFF_1:26, SIN_COS:54, TAYLOR_1:16;
then A7: (arctan + arccot) / exp_R is_differentiable_on Z by A2, FDIFF_2:21;
 for x being Real st x in Z holds
(((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x))
proof
let x be Real; ::_thesis: ( x in Z implies (((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) )
A8: exp_R is_differentiable_in x by SIN_COS:65;
A9: exp_R . x <> 0 by SIN_COS:54;
assume A10: x in Z ; ::_thesis: (((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x))
then A11: arctan + arccot is_differentiable_in x by A2, FDIFF_1:9;
(((arctan + arccot) / exp_R) `| Z) . x = diff (((arctan + arccot) / exp_R),x) by A7, A10, FDIFF_1:def_7
.= (((diff ((arctan + arccot),x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan + arccot) . x))) / ((exp_R . x) ^2) by A11, A8, A9, FDIFF_2:14
.= (((((arctan + arccot) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan + arccot) . x))) / ((exp_R . x) ^2) by A2, A10, FDIFF_1:def_7
.= ((0 * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan + arccot) . x))) / ((exp_R . x) ^2) by A1, A10, Th37
.= - (((diff (exp_R,x)) * ((arctan + arccot) . x)) / ((exp_R . x) ^2))
.= - (((exp_R . x) * ((arctan + arccot) . x)) / ((exp_R . x) ^2)) by SIN_COS:65
.= - ((((arctan . x) + (arccot . x)) * (exp_R . x)) / ((exp_R . x) * (exp_R . x))) by A5, A10, VALUED_1:def_1
.= - (((arctan . x) + (arccot . x)) * ((exp_R . x) / ((exp_R . x) * (exp_R . x))))
.= - (((arctan . x) + (arccot . x)) * (((exp_R . x) / (exp_R . x)) / (exp_R . x))) by XCMPLX_1:78
.= - (((arctan . x) + (arccot . x)) * (1 / (exp_R . x))) by A9, XCMPLX_1:60
.= - (((arctan . x) + (arccot . x)) / (exp_R . x)) ;
hence  (((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ; ::_thesis: verum
end;
hence  ( (arctan + arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ) ) by A2, A6, FDIFF_2:21; ::_thesis: verum
end;

theorem :: FDIFF_11:54
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( (arctan - arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( (arctan - arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ) ) )
assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( (arctan - arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ) )
then A2: arctan - arccot is_differentiable_on Z by Th38;
A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A4: Z c= dom arccot by A1, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A1, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19;
then A5: Z c= dom (arctan - arccot) by VALUED_1:12;
A6: ( exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
exp_R . x <> 0 ) ) by FDIFF_1:26, SIN_COS:54, TAYLOR_1:16;
then A7: (arctan - arccot) / exp_R is_differentiable_on Z by A2, FDIFF_2:21;
 for x being Real st x in Z holds
(((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x)
proof
let x be Real; ::_thesis: ( x in Z implies (((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) )
A8: exp_R is_differentiable_in x by SIN_COS:65;
A9: exp_R . x <> 0 by SIN_COS:54;
assume A10: x in Z ; ::_thesis: (((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x)
then A11: arctan - arccot is_differentiable_in x by A2, FDIFF_1:9;
(((arctan - arccot) / exp_R) `| Z) . x = diff (((arctan - arccot) / exp_R),x) by A7, A10, FDIFF_1:def_7
.= (((diff ((arctan - arccot),x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x))) / ((exp_R . x) ^2) by A11, A8, A9, FDIFF_2:14
.= (((((arctan - arccot) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x))) / ((exp_R . x) ^2) by A2, A10, FDIFF_1:def_7
.= (((2 / (1 + (x ^2))) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x))) / ((exp_R . x) ^2) by A1, A10, Th38
.= (((2 / (1 + (x ^2))) * (exp_R . x)) - ((exp_R . x) * ((arctan - arccot) . x))) / ((exp_R . x) ^2) by SIN_COS:65
.= (((2 / (1 + (x ^2))) * (exp_R . x)) - ((exp_R . x) * ((arctan . x) - (arccot . x)))) / ((exp_R . x) ^2) by A5, A10, VALUED_1:13
.= ((2 / (1 + (x ^2))) - ((arctan . x) - (arccot . x))) * ((exp_R . x) / ((exp_R . x) * (exp_R . x)))
.= ((2 / (1 + (x ^2))) - ((arctan . x) - (arccot . x))) * (((exp_R . x) / (exp_R . x)) / (exp_R . x)) by XCMPLX_1:78
.= ((2 / (1 + (x ^2))) - ((arctan . x) - (arccot . x))) * (1 / (exp_R . x)) by A9, XCMPLX_1:60
.= (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ;
hence  (((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ; ::_thesis: verum
end;
hence  ( (arctan - arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ) ) by A2, A6, FDIFF_2:21; ::_thesis: verum
end;

theorem :: FDIFF_11:55
for Z being open Subset of REAL st Z c= dom (exp_R * (arctan + arccot)) & Z c= ].(- 1),1.[ holds
( exp_R * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * (arctan + arccot)) `| Z) . x = 0 ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (exp_R * (arctan + arccot)) & Z c= ].(- 1),1.[ implies ( exp_R * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * (arctan + arccot)) `| Z) . x = 0 ) ) )
assume that
A1: Z c= dom (exp_R * (arctan + arccot)) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( exp_R * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * (arctan + arccot)) `| Z) . x = 0 ) )
A3: arctan + arccot is_differentiable_on Z by A2, Th37;
A4: for x being Real st x in Z holds
exp_R * (arctan + arccot) is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies exp_R * (arctan + arccot) is_differentiable_in x )
assume  x in Z ; ::_thesis: exp_R * (arctan + arccot) is_differentiable_in x
then A5: arctan + arccot is_differentiable_in x by A3, FDIFF_1:9;
 exp_R is_differentiable_in (arctan + arccot) . x by SIN_COS:65;
hence  exp_R * (arctan + arccot) is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum
end;
then A6: exp_R * (arctan + arccot) is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((exp_R * (arctan + arccot)) `| Z) . x = 0
proof
let x be Real; ::_thesis: ( x in Z implies ((exp_R * (arctan + arccot)) `| Z) . x = 0 )
A7: exp_R is_differentiable_in (arctan + arccot) . x by SIN_COS:65;
assume A8: x in Z ; ::_thesis: ((exp_R * (arctan + arccot)) `| Z) . x = 0
then A9: arctan + arccot is_differentiable_in x by A3, FDIFF_1:9;
((exp_R * (arctan + arccot)) `| Z) . x = diff ((exp_R * (arctan + arccot)),x) by A6, A8, FDIFF_1:def_7
.= (diff (exp_R,((arctan + arccot) . x))) * (diff ((arctan + arccot),x)) by A9, A7, FDIFF_2:13
.= (exp_R . ((arctan + arccot) . x)) * (diff ((arctan + arccot),x)) by SIN_COS:65
.= (exp_R . ((arctan + arccot) . x)) * (((arctan + arccot) `| Z) . x) by A3, A8, FDIFF_1:def_7
.= (exp_R . ((arctan + arccot) . x)) * 0 by A2, A8, Th37
.= 0 ;
hence  ((exp_R * (arctan + arccot)) `| Z) . x = 0 ; ::_thesis: verum
end;
hence  ( exp_R * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * (arctan + arccot)) `| Z) . x = 0 ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:56
for Z being open Subset of REAL st Z c= dom (exp_R * (arctan - arccot)) & Z c= ].(- 1),1.[ holds
( exp_R * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * (arctan - arccot)) `| Z) . x = (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (exp_R * (arctan - arccot)) & Z c= ].(- 1),1.[ implies ( exp_R * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * (arctan - arccot)) `| Z) . x = (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) ) )
assume that
A1: Z c= dom (exp_R * (arctan - arccot)) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( exp_R * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * (arctan - arccot)) `| Z) . x = (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) )
A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A4: Z c= dom arccot by A2, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A2, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19;
then A5: Z c= dom (arctan - arccot) by VALUED_1:12;
A6: arctan - arccot is_differentiable_on Z by A2, Th38;
A7: for x being Real st x in Z holds
exp_R * (arctan - arccot) is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies exp_R * (arctan - arccot) is_differentiable_in x )
assume  x in Z ; ::_thesis: exp_R * (arctan - arccot) is_differentiable_in x
then A8: arctan - arccot is_differentiable_in x by A6, FDIFF_1:9;
 exp_R is_differentiable_in (arctan - arccot) . x by SIN_COS:65;
hence  exp_R * (arctan - arccot) is_differentiable_in x by A8, FDIFF_2:13; ::_thesis: verum
end;
then A9: exp_R * (arctan - arccot) is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((exp_R * (arctan - arccot)) `| Z) . x = (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))
proof
let x be Real; ::_thesis: ( x in Z implies ((exp_R * (arctan - arccot)) `| Z) . x = (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) )
A10: exp_R is_differentiable_in (arctan - arccot) . x by SIN_COS:65;
assume A11: x in Z ; ::_thesis: ((exp_R * (arctan - arccot)) `| Z) . x = (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))
then A12: arctan - arccot is_differentiable_in x by A6, FDIFF_1:9;
((exp_R * (arctan - arccot)) `| Z) . x = diff ((exp_R * (arctan - arccot)),x) by A9, A11, FDIFF_1:def_7
.= (diff (exp_R,((arctan - arccot) . x))) * (diff ((arctan - arccot),x)) by A12, A10, FDIFF_2:13
.= (exp_R . ((arctan - arccot) . x)) * (diff ((arctan - arccot),x)) by SIN_COS:65
.= (exp_R . ((arctan - arccot) . x)) * (((arctan - arccot) `| Z) . x) by A6, A11, FDIFF_1:def_7
.= (exp_R . ((arctan - arccot) . x)) * (2 / (1 + (x ^2))) by A2, A11, Th38
.= (exp_R . ((arctan . x) - (arccot . x))) * (2 / (1 + (x ^2))) by A5, A11, VALUED_1:13
.= (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ;
hence  ((exp_R * (arctan - arccot)) `| Z) . x = (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ; ::_thesis: verum
end;
hence  ( exp_R * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * (arctan - arccot)) `| Z) . x = (2 * (exp_R . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) ) by A1, A7, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:57
for Z being open Subset of REAL st Z c= dom (sin * (arctan + arccot)) & Z c= ].(- 1),1.[ holds
( sin * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * (arctan + arccot)) `| Z) . x = 0 ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sin * (arctan + arccot)) & Z c= ].(- 1),1.[ implies ( sin * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * (arctan + arccot)) `| Z) . x = 0 ) ) )
assume that
A1: Z c= dom (sin * (arctan + arccot)) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sin * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * (arctan + arccot)) `| Z) . x = 0 ) )
A3: arctan + arccot is_differentiable_on Z by A2, Th37;
A4: for x being Real st x in Z holds
sin * (arctan + arccot) is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies sin * (arctan + arccot) is_differentiable_in x )
assume  x in Z ; ::_thesis: sin * (arctan + arccot) is_differentiable_in x
then A5: arctan + arccot is_differentiable_in x by A3, FDIFF_1:9;
 sin is_differentiable_in (arctan + arccot) . x by SIN_COS:64;
hence  sin * (arctan + arccot) is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum
end;
then A6: sin * (arctan + arccot) is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((sin * (arctan + arccot)) `| Z) . x = 0
proof
let x be Real; ::_thesis: ( x in Z implies ((sin * (arctan + arccot)) `| Z) . x = 0 )
A7: sin is_differentiable_in (arctan + arccot) . x by SIN_COS:64;
assume A8: x in Z ; ::_thesis: ((sin * (arctan + arccot)) `| Z) . x = 0
then A9: arctan + arccot is_differentiable_in x by A3, FDIFF_1:9;
((sin * (arctan + arccot)) `| Z) . x = diff ((sin * (arctan + arccot)),x) by A6, A8, FDIFF_1:def_7
.= (diff (sin,((arctan + arccot) . x))) * (diff ((arctan + arccot),x)) by A9, A7, FDIFF_2:13
.= (cos . ((arctan + arccot) . x)) * (diff ((arctan + arccot),x)) by SIN_COS:64
.= (cos . ((arctan + arccot) . x)) * (((arctan + arccot) `| Z) . x) by A3, A8, FDIFF_1:def_7
.= (cos . ((arctan + arccot) . x)) * 0 by A2, A8, Th37
.= 0 ;
hence  ((sin * (arctan + arccot)) `| Z) . x = 0 ; ::_thesis: verum
end;
hence  ( sin * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * (arctan + arccot)) `| Z) . x = 0 ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:58
for Z being open Subset of REAL st Z c= dom (sin * (arctan - arccot)) & Z c= ].(- 1),1.[ holds
( sin * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * (arctan - arccot)) `| Z) . x = (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (sin * (arctan - arccot)) & Z c= ].(- 1),1.[ implies ( sin * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * (arctan - arccot)) `| Z) . x = (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) ) )
assume that
A1: Z c= dom (sin * (arctan - arccot)) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( sin * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * (arctan - arccot)) `| Z) . x = (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) )
A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A4: Z c= dom arccot by A2, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A2, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19;
then A5: Z c= dom (arctan - arccot) by VALUED_1:12;
A6: arctan - arccot is_differentiable_on Z by A2, Th38;
A7: for x being Real st x in Z holds
sin * (arctan - arccot) is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies sin * (arctan - arccot) is_differentiable_in x )
assume  x in Z ; ::_thesis: sin * (arctan - arccot) is_differentiable_in x
then A8: arctan - arccot is_differentiable_in x by A6, FDIFF_1:9;
 sin is_differentiable_in (arctan - arccot) . x by SIN_COS:64;
hence  sin * (arctan - arccot) is_differentiable_in x by A8, FDIFF_2:13; ::_thesis: verum
end;
then A9: sin * (arctan - arccot) is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((sin * (arctan - arccot)) `| Z) . x = (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))
proof
let x be Real; ::_thesis: ( x in Z implies ((sin * (arctan - arccot)) `| Z) . x = (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) )
A10: sin is_differentiable_in (arctan - arccot) . x by SIN_COS:64;
assume A11: x in Z ; ::_thesis: ((sin * (arctan - arccot)) `| Z) . x = (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))
then A12: arctan - arccot is_differentiable_in x by A6, FDIFF_1:9;
((sin * (arctan - arccot)) `| Z) . x = diff ((sin * (arctan - arccot)),x) by A9, A11, FDIFF_1:def_7
.= (diff (sin,((arctan - arccot) . x))) * (diff ((arctan - arccot),x)) by A12, A10, FDIFF_2:13
.= (cos . ((arctan - arccot) . x)) * (diff ((arctan - arccot),x)) by SIN_COS:64
.= (cos . ((arctan - arccot) . x)) * (((arctan - arccot) `| Z) . x) by A6, A11, FDIFF_1:def_7
.= (cos . ((arctan - arccot) . x)) * (2 / (1 + (x ^2))) by A2, A11, Th38
.= (cos . ((arctan . x) - (arccot . x))) * (2 / (1 + (x ^2))) by A5, A11, VALUED_1:13
.= (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ;
hence  ((sin * (arctan - arccot)) `| Z) . x = (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ; ::_thesis: verum
end;
hence  ( sin * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * (arctan - arccot)) `| Z) . x = (2 * (cos . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)) ) ) by A1, A7, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:59
for Z being open Subset of REAL st Z c= dom (cos * (arctan + arccot)) & Z c= ].(- 1),1.[ holds
( cos * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (arctan + arccot)) `| Z) . x = 0 ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cos * (arctan + arccot)) & Z c= ].(- 1),1.[ implies ( cos * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (arctan + arccot)) `| Z) . x = 0 ) ) )
assume that
A1: Z c= dom (cos * (arctan + arccot)) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cos * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (arctan + arccot)) `| Z) . x = 0 ) )
A3: arctan + arccot is_differentiable_on Z by A2, Th37;
A4: for x being Real st x in Z holds
cos * (arctan + arccot) is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cos * (arctan + arccot) is_differentiable_in x )
assume  x in Z ; ::_thesis: cos * (arctan + arccot) is_differentiable_in x
then A5: arctan + arccot is_differentiable_in x by A3, FDIFF_1:9;
 cos is_differentiable_in (arctan + arccot) . x by SIN_COS:63;
hence  cos * (arctan + arccot) is_differentiable_in x by A5, FDIFF_2:13; ::_thesis: verum
end;
then A6: cos * (arctan + arccot) is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((cos * (arctan + arccot)) `| Z) . x = 0
proof
let x be Real; ::_thesis: ( x in Z implies ((cos * (arctan + arccot)) `| Z) . x = 0 )
A7: cos is_differentiable_in (arctan + arccot) . x by SIN_COS:63;
assume A8: x in Z ; ::_thesis: ((cos * (arctan + arccot)) `| Z) . x = 0
then A9: arctan + arccot is_differentiable_in x by A3, FDIFF_1:9;
((cos * (arctan + arccot)) `| Z) . x = diff ((cos * (arctan + arccot)),x) by A6, A8, FDIFF_1:def_7
.= (diff (cos,((arctan + arccot) . x))) * (diff ((arctan + arccot),x)) by A9, A7, FDIFF_2:13
.= (- (sin . ((arctan + arccot) . x))) * (diff ((arctan + arccot),x)) by SIN_COS:63
.= (- (sin . ((arctan + arccot) . x))) * (((arctan + arccot) `| Z) . x) by A3, A8, FDIFF_1:def_7
.= (- (sin . ((arctan + arccot) . x))) * 0 by A2, A8, Th37
.= 0 ;
hence  ((cos * (arctan + arccot)) `| Z) . x = 0 ; ::_thesis: verum
end;
hence  ( cos * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (arctan + arccot)) `| Z) . x = 0 ) ) by A1, A4, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:60
for Z being open Subset of REAL st Z c= dom (cos * (arctan - arccot)) & Z c= ].(- 1),1.[ holds
( cos * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (arctan - arccot)) `| Z) . x = - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (cos * (arctan - arccot)) & Z c= ].(- 1),1.[ implies ( cos * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (arctan - arccot)) `| Z) . x = - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (cos * (arctan - arccot)) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( cos * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (arctan - arccot)) `| Z) . x = - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) ) )
A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A4: Z c= dom arccot by A2, XBOOLE_1:1;
 ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A2, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19;
then A5: Z c= dom (arctan - arccot) by VALUED_1:12;
A6: arctan - arccot is_differentiable_on Z by A2, Th38;
A7: for x being Real st x in Z holds
cos * (arctan - arccot) is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies cos * (arctan - arccot) is_differentiable_in x )
assume  x in Z ; ::_thesis: cos * (arctan - arccot) is_differentiable_in x
then A8: arctan - arccot is_differentiable_in x by A6, FDIFF_1:9;
 cos is_differentiable_in (arctan - arccot) . x by SIN_COS:63;
hence  cos * (arctan - arccot) is_differentiable_in x by A8, FDIFF_2:13; ::_thesis: verum
end;
then A9: cos * (arctan - arccot) is_differentiable_on Z by A1, FDIFF_1:9;
 for x being Real st x in Z holds
((cos * (arctan - arccot)) `| Z) . x = - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((cos * (arctan - arccot)) `| Z) . x = - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) )
A10: cos is_differentiable_in (arctan - arccot) . x by SIN_COS:63;
assume A11: x in Z ; ::_thesis: ((cos * (arctan - arccot)) `| Z) . x = - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2)))
then A12: arctan - arccot is_differentiable_in x by A6, FDIFF_1:9;
((cos * (arctan - arccot)) `| Z) . x = diff ((cos * (arctan - arccot)),x) by A9, A11, FDIFF_1:def_7
.= (diff (cos,((arctan - arccot) . x))) * (diff ((arctan - arccot),x)) by A12, A10, FDIFF_2:13
.= (- (sin . ((arctan - arccot) . x))) * (diff ((arctan - arccot),x)) by SIN_COS:63
.= (- (sin . ((arctan - arccot) . x))) * (((arctan - arccot) `| Z) . x) by A6, A11, FDIFF_1:def_7
.= (- (sin . ((arctan - arccot) . x))) * (2 / (1 + (x ^2))) by A2, A11, Th38
.= (- (sin . ((arctan . x) - (arccot . x)))) * (2 / (1 + (x ^2))) by A5, A11, VALUED_1:13
.= - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) ;
hence  ((cos * (arctan - arccot)) `| Z) . x = - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( cos * (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (arctan - arccot)) `| Z) . x = - ((2 * (sin . ((arctan . x) - (arccot . x)))) / (1 + (x ^2))) ) ) by A1, A7, FDIFF_1:9; ::_thesis: verum
end;

theorem :: FDIFF_11:61
for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( arctan (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ implies ( arctan (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2)) ) ) )
assume A1: Z c= ].(- 1),1.[ ; ::_thesis: ( arctan (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2)) ) )
then A2: arctan is_differentiable_on Z by SIN_COS9:81;
A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A4: Z c= dom arccot by A1, XBOOLE_1:1;
A5: arccot is_differentiable_on Z by A1, SIN_COS9:82;
 ].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1;
then  Z c= dom arctan by A1, XBOOLE_1:1;
then  Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19;
then A6: Z c= dom (arctan (#) arccot) by VALUED_1:def_4;
 for x being Real st x in Z holds
((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2))
proof
let x be Real; ::_thesis: ( x in Z implies ((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2)) )
assume A7: x in Z ; ::_thesis: ((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2))
then ((arctan (#) arccot) `| Z) . x = ((arccot . x) * (diff (arctan,x))) + ((arctan . x) * (diff (arccot,x))) by A6, A2, A5, FDIFF_1:21
.= ((arccot . x) * ((arctan `| Z) . x)) + ((arctan . x) * (diff (arccot,x))) by A2, A7, FDIFF_1:def_7
.= ((arccot . x) * (1 / (1 + (x ^2)))) + ((arctan . x) * (diff (arccot,x))) by A1, A7, SIN_COS9:81
.= ((arccot . x) * (1 / (1 + (x ^2)))) + ((arctan . x) * ((arccot `| Z) . x)) by A5, A7, FDIFF_1:def_7
.= ((arccot . x) * (1 / (1 + (x ^2)))) + ((arctan . x) * (- (1 / (1 + (x ^2))))) by A1, A7, SIN_COS9:82
.= ((arccot . x) - (arctan . x)) / (1 + (x ^2)) ;
hence  ((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2)) ; ::_thesis: verum
end;
hence  ( arctan (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2)) ) ) by A6, A2, A5, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:62
for Z being open Subset of REAL st not 0 in Z & Z c= dom ((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds
( (arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( not 0 in Z & Z c= dom ((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) implies ( (arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) ) ) )
set f = id Z;
assume that
A1: not 0 in Z and
A2: Z c= dom ((arctan * ((id Z) ^)) (#) (arccot * ((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) ^)) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) ) )
A4: Z c= (dom (arctan * ((id Z) ^))) /\ (dom (arccot * ((id Z) ^))) by A2, VALUED_1:def_4;
then A5: Z c= dom (arctan * ((id Z) ^)) by XBOOLE_1:18;
then A6: arctan * ((id Z) ^) is_differentiable_on Z by A1, A3, SIN_COS9:111;
A7: Z c= dom (arccot * ((id Z) ^)) by A4, XBOOLE_1:18;
then A8: arccot * ((id Z) ^) is_differentiable_on Z by A1, A3, SIN_COS9:112;
 for y being set st y in Z holds
y in dom ((id Z) ^) by A5, FUNCT_1:11;
then A9: Z c= dom ((id Z) ^) by TARSKI:def_3;
 for x being Real st x in Z holds
(((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2))
proof
let x be Real; ::_thesis: ( x in Z implies (((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) )
assume A10: x in Z ; ::_thesis: (((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2))
then (((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = (((arccot * ((id Z) ^)) . x) * (diff ((arctan * ((id Z) ^)),x))) + (((arctan * ((id Z) ^)) . x) * (diff ((arccot * ((id Z) ^)),x))) by A2, A6, A8, FDIFF_1:21
.= (((arccot * ((id Z) ^)) . x) * (((arctan * ((id Z) ^)) `| Z) . x)) + (((arctan * ((id Z) ^)) . x) * (diff ((arccot * ((id Z) ^)),x))) by A6, A10, FDIFF_1:def_7
.= (((arccot * ((id Z) ^)) . x) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (diff ((arccot * ((id Z) ^)),x))) by A1, A3, A5, A10, SIN_COS9:111
.= (((arccot * ((id Z) ^)) . x) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (((arccot * ((id Z) ^)) `| Z) . x)) by A8, A10, FDIFF_1:def_7
.= (((arccot * ((id Z) ^)) . x) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (1 / (1 + (x ^2)))) by A1, A3, A7, A10, SIN_COS9:112
.= ((arccot . (((id Z) ^) . x)) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (1 / (1 + (x ^2)))) by A7, A10, FUNCT_1:12
.= ((arccot . (((id Z) . x) ")) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (1 / (1 + (x ^2)))) by A9, A10, RFUNCT_1:def_2
.= ((arccot . (1 / x)) * (- (1 / (1 + (x ^2))))) + (((arctan * ((id Z) ^)) . x) * (1 / (1 + (x ^2)))) by A10, FUNCT_1:18
.= ((arccot . (1 / x)) * (- (1 / (1 + (x ^2))))) + ((arctan . (((id Z) ^) . x)) * (1 / (1 + (x ^2)))) by A5, A10, FUNCT_1:12
.= ((arccot . (1 / x)) * (- (1 / (1 + (x ^2))))) + ((arctan . (((id Z) . x) ")) * (1 / (1 + (x ^2)))) by A9, A10, RFUNCT_1:def_2
.= (- ((arccot . (1 / x)) * (1 / (1 + (x ^2))))) + ((arctan . (1 / x)) * (1 / (1 + (x ^2)))) by A10, FUNCT_1:18
.= ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) ;
hence  (((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) ; ::_thesis: verum
end;
hence  ( (arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan * ((id Z) ^)) (#) (arccot * ((id Z) ^))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2)) ) ) by A2, A6, A8, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:63
for Z being open Subset of REAL st not 0 in Z & Z c= dom ((id Z) (#) (arctan * ((id Z) ^))) & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds
( (id Z) (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( not 0 in Z & Z c= dom ((id Z) (#) (arctan * ((id Z) ^))) & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) implies ( (id Z) (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) ) ) )
set f = id Z;
assume that
A1: not 0 in Z and
A2: Z c= dom ((id Z) (#) (arctan * ((id Z) ^))) and
A3: for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ; ::_thesis: ( (id Z) (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) ) )
A4: Z c= (dom (id Z)) /\ (dom (arctan * ((id Z) ^))) by A2, VALUED_1:def_4;
then A5: Z c= dom (arctan * ((id Z) ^)) by XBOOLE_1:18;
then A6: arctan * ((id Z) ^) is_differentiable_on Z by A1, A3, SIN_COS9:111;
A7: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
A8: Z c= dom (id Z) by A4, XBOOLE_1:18;
then A9: id Z is_differentiable_on Z by A7, FDIFF_1:23;
 for y being set st y in Z holds
y in dom ((id Z) ^) by A5, FUNCT_1:11;
then A10: Z c= dom ((id Z) ^) by TARSKI:def_3;
 for x being Real st x in Z holds
(((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies (((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) )
assume A11: x in Z ; ::_thesis: (((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2)))
then (((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (((arctan * ((id Z) ^)) . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff ((arctan * ((id Z) ^)),x))) by A2, A6, A9, FDIFF_1:21
.= (((arctan * ((id Z) ^)) . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff ((arctan * ((id Z) ^)),x))) by A9, A11, FDIFF_1:def_7
.= (((arctan * ((id Z) ^)) . x) * 1) + (((id Z) . x) * (diff ((arctan * ((id Z) ^)),x))) by A8, A7, A11, FDIFF_1:23
.= (((arctan * ((id Z) ^)) . x) * 1) + (x * (diff ((arctan * ((id Z) ^)),x))) by A11, FUNCT_1:18
.= ((arctan * ((id Z) ^)) . x) + (x * (((arctan * ((id Z) ^)) `| Z) . x)) by A6, A11, FDIFF_1:def_7
.= ((arctan * ((id Z) ^)) . x) + (x * (- (1 / (1 + (x ^2))))) by A1, A3, A5, A11, SIN_COS9:111
.= (arctan . (((id Z) ^) . x)) - (x / (1 + (x ^2))) by A5, A11, FUNCT_1:12
.= (arctan . (((id Z) . x) ")) - (x / (1 + (x ^2))) by A10, A11, RFUNCT_1:def_2
.= (arctan . (1 / x)) - (x / (1 + (x ^2))) by A11, FUNCT_1:18 ;
hence  (((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( (id Z) (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) ) ) by A2, A6, A9, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:64
for Z being open Subset of REAL st not 0 in Z & Z c= dom ((id Z) (#) (arccot * ((id Z) ^))) & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds
( (id Z) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (arccot . (1 / x)) + (x / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( not 0 in Z & Z c= dom ((id Z) (#) (arccot * ((id Z) ^))) & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) implies ( (id Z) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (arccot . (1 / x)) + (x / (1 + (x ^2))) ) ) )
set f = id Z;
assume that
A1: not 0 in Z and
A2: Z c= dom ((id Z) (#) (arccot * ((id Z) ^))) and
A3: for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ; ::_thesis: ( (id Z) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (arccot . (1 / x)) + (x / (1 + (x ^2))) ) )
A4: Z c= (dom (id Z)) /\ (dom (arccot * ((id Z) ^))) by A2, VALUED_1:def_4;
then A5: Z c= dom (arccot * ((id Z) ^)) by XBOOLE_1:18;
then A6: arccot * ((id Z) ^) is_differentiable_on Z by A1, A3, SIN_COS9:112;
A7: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
A8: Z c= dom (id Z) by A4, XBOOLE_1:18;
then A9: id Z is_differentiable_on Z by A7, FDIFF_1:23;
 for y being set st y in Z holds
y in dom ((id Z) ^) by A5, FUNCT_1:11;
then A10: Z c= dom ((id Z) ^) by TARSKI:def_3;
 for x being Real st x in Z holds
(((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (arccot . (1 / x)) + (x / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies (((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (arccot . (1 / x)) + (x / (1 + (x ^2))) )
assume A11: x in Z ; ::_thesis: (((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (arccot . (1 / x)) + (x / (1 + (x ^2)))
then (((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (((arccot * ((id Z) ^)) . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff ((arccot * ((id Z) ^)),x))) by A2, A6, A9, FDIFF_1:21
.= (((arccot * ((id Z) ^)) . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff ((arccot * ((id Z) ^)),x))) by A9, A11, FDIFF_1:def_7
.= (((arccot * ((id Z) ^)) . x) * 1) + (((id Z) . x) * (diff ((arccot * ((id Z) ^)),x))) by A8, A7, A11, FDIFF_1:23
.= (((arccot * ((id Z) ^)) . x) * 1) + (x * (diff ((arccot * ((id Z) ^)),x))) by A11, FUNCT_1:18
.= ((arccot * ((id Z) ^)) . x) + (x * (((arccot * ((id Z) ^)) `| Z) . x)) by A6, A11, FDIFF_1:def_7
.= ((arccot * ((id Z) ^)) . x) + (x * (1 / (1 + (x ^2)))) by A1, A3, A5, A11, SIN_COS9:112
.= (arccot . (((id Z) ^) . x)) + (x / (1 + (x ^2))) by A5, A11, FUNCT_1:12
.= (arccot . (((id Z) . x) ")) + (x / (1 + (x ^2))) by A10, A11, RFUNCT_1:def_2
.= (arccot . (1 / x)) + (x / (1 + (x ^2))) by A11, FUNCT_1:18 ;
hence  (((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (arccot . (1 / x)) + (x / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( (id Z) (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arccot * ((id Z) ^))) `| Z) . x = (arccot . (1 / x)) + (x / (1 + (x ^2))) ) ) by A2, A6, A9, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:65
for Z being open Subset of REAL
for g being PartFunc of REAL,REAL st not 0 in Z & Z c= dom (g (#) (arctan * ((id Z) ^))) & g = #Z 2 & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds
( g (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: for g being PartFunc of REAL,REAL st not 0 in Z & Z c= dom (g (#) (arctan * ((id Z) ^))) & g = #Z 2 & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds
( g (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) ) )
let g be PartFunc of REAL,REAL; ::_thesis: ( not 0 in Z & Z c= dom (g (#) (arctan * ((id Z) ^))) & g = #Z 2 & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) implies ( g (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) ) ) )
set f = id Z;
assume that
A1: not 0 in Z and
A2: Z c= dom (g (#) (arctan * ((id Z) ^))) and
A3: g = #Z 2 and
A4: for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ; ::_thesis: ( g (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) ) )
A5: for x being Real st x in Z holds
g is_differentiable_in x by A3, TAYLOR_1:2;
A6: Z c= (dom g) /\ (dom (arctan * ((id Z) ^))) by A2, VALUED_1:def_4;
then A7: Z c= dom (arctan * ((id Z) ^)) by XBOOLE_1:18;
then A8: arctan * ((id Z) ^) is_differentiable_on Z by A1, A4, SIN_COS9:111;
 Z c= dom g by A6, XBOOLE_1:18;
then A9: g is_differentiable_on Z by A5, FDIFF_1:9;
A10: for x being Real st x in Z holds
(g `| Z) . x = 2 * x
proof
let x be Real; ::_thesis: ( x in Z implies (g `| Z) . x = 2 * x )
assume  x in Z ; ::_thesis: (g `| Z) . x = 2 * x
then (g `| Z) . x = diff (g,x) by A9, FDIFF_1:def_7
.= 2 * (x #Z (2 - 1)) by A3, TAYLOR_1:2
.= 2 * x by PREPOWER:35 ;
hence  (g `| Z) . x = 2 * x ; ::_thesis: verum
end;
 for y being set st y in Z holds
y in dom ((id Z) ^) by A7, FUNCT_1:11;
then A11: Z c= dom ((id Z) ^) by TARSKI:def_3;
 for x being Real st x in Z holds
((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) )
assume A12: x in Z ; ::_thesis: ((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2)))
then ((g (#) (arctan * ((id Z) ^))) `| Z) . x = (((arctan * ((id Z) ^)) . x) * (diff (g,x))) + ((g . x) * (diff ((arctan * ((id Z) ^)),x))) by A2, A8, A9, FDIFF_1:21
.= (((arctan * ((id Z) ^)) . x) * ((g `| Z) . x)) + ((g . x) * (diff ((arctan * ((id Z) ^)),x))) by A9, A12, FDIFF_1:def_7
.= (((arctan * ((id Z) ^)) . x) * (2 * x)) + ((g . x) * (diff ((arctan * ((id Z) ^)),x))) by A10, A12
.= (((arctan * ((id Z) ^)) . x) * (2 * x)) + ((x #Z 2) * (diff ((arctan * ((id Z) ^)),x))) by A3, TAYLOR_1:def_1
.= (((arctan * ((id Z) ^)) . x) * (2 * x)) + ((x #Z 2) * (((arctan * ((id Z) ^)) `| Z) . x)) by A8, A12, FDIFF_1:def_7
.= (((arctan * ((id Z) ^)) . x) * (2 * x)) + ((x #Z (1 + 1)) * (- (1 / (1 + (x ^2))))) by A1, A4, A7, A12, SIN_COS9:111
.= (((arctan * ((id Z) ^)) . x) * (2 * x)) + (((x #Z 1) * (x #Z 1)) * (- (1 / (1 + (x ^2))))) by TAYLOR_1:1
.= (((arctan * ((id Z) ^)) . x) * (2 * x)) + ((x * (x #Z 1)) * (- (1 / (1 + (x ^2))))) by PREPOWER:35
.= (((arctan * ((id Z) ^)) . x) * (2 * x)) + ((x ^2) * (- (1 / (1 + (x ^2))))) by PREPOWER:35
.= ((arctan . (((id Z) ^) . x)) * (2 * x)) - ((x ^2) / (1 + (x ^2))) by A7, A12, FUNCT_1:12
.= ((arctan . (((id Z) . x) ")) * (2 * x)) - ((x ^2) / (1 + (x ^2))) by A11, A12, RFUNCT_1:def_2
.= ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) by A12, FUNCT_1:18 ;
hence  ((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( g (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
((g (#) (arctan * ((id Z) ^))) `| Z) . x = ((2 * x) * (arctan . (1 / x))) - ((x ^2) / (1 + (x ^2))) ) ) by A2, A8, A9, FDIFF_1:21; ::_thesis: verum
end;

theorem :: FDIFF_11:66
for Z being open Subset of REAL
for g being PartFunc of REAL,REAL st not 0 in Z & Z c= dom (g (#) (arccot * ((id Z) ^))) & g = #Z 2 & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds
( g (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: for g being PartFunc of REAL,REAL st not 0 in Z & Z c= dom (g (#) (arccot * ((id Z) ^))) & g = #Z 2 & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) holds
( g (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) ) )
let g be PartFunc of REAL,REAL; ::_thesis: ( not 0 in Z & Z c= dom (g (#) (arccot * ((id Z) ^))) & g = #Z 2 & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) implies ( g (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) ) ) )
set f = id Z;
assume that
A1: not 0 in Z and
A2: Z c= dom (g (#) (arccot * ((id Z) ^))) and
A3: g = #Z 2 and
A4: for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ; ::_thesis: ( g (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) ) )
A5: for x being Real st x in Z holds
g is_differentiable_in x by A3, TAYLOR_1:2;
A6: Z c= (dom g) /\ (dom (arccot * ((id Z) ^))) by A2, VALUED_1:def_4;
then A7: Z c= dom (arccot * ((id Z) ^)) by XBOOLE_1:18;
then A8: arccot * ((id Z) ^) is_differentiable_on Z by A1, A4, SIN_COS9:112;
 Z c= dom g by A6, XBOOLE_1:18;
then A9: g is_differentiable_on Z by A5, FDIFF_1:9;
A10: for x being Real st x in Z holds
(g `| Z) . x = 2 * x
proof
let x be Real; ::_thesis: ( x in Z implies (g `| Z) . x = 2 * x )
assume  x in Z ; ::_thesis: (g `| Z) . x = 2 * x
then (g `| Z) . x = diff (g,x) by A9, FDIFF_1:def_7
.= 2 * (x #Z (2 - 1)) by A3, TAYLOR_1:2
.= 2 * x by PREPOWER:35 ;
hence  (g `| Z) . x = 2 * x ; ::_thesis: verum
end;
 for y being set st y in Z holds
y in dom ((id Z) ^) by A7, FUNCT_1:11;
then A11: Z c= dom ((id Z) ^) by TARSKI:def_3;
 for x being Real st x in Z holds
((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) )
assume A12: x in Z ; ::_thesis: ((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2)))
then ((g (#) (arccot * ((id Z) ^))) `| Z) . x = (((arccot * ((id Z) ^)) . x) * (diff (g,x))) + ((g . x) * (diff ((arccot * ((id Z) ^)),x))) by A2, A8, A9, FDIFF_1:21
.= (((arccot * ((id Z) ^)) . x) * ((g `| Z) . x)) + ((g . x) * (diff ((arccot * ((id Z) ^)),x))) by A9, A12, FDIFF_1:def_7
.= (((arccot * ((id Z) ^)) . x) * (2 * x)) + ((g . x) * (diff ((arccot * ((id Z) ^)),x))) by A10, A12
.= (((arccot * ((id Z) ^)) . x) * (2 * x)) + ((x #Z 2) * (diff ((arccot * ((id Z) ^)),x))) by A3, TAYLOR_1:def_1
.= (((arccot * ((id Z) ^)) . x) * (2 * x)) + ((x #Z 2) * (((arccot * ((id Z) ^)) `| Z) . x)) by A8, A12, FDIFF_1:def_7
.= (((arccot * ((id Z) ^)) . x) * (2 * x)) + ((x #Z (1 + 1)) * (1 / (1 + (x ^2)))) by A1, A4, A7, A12, SIN_COS9:112
.= (((arccot * ((id Z) ^)) . x) * (2 * x)) + (((x #Z 1) * (x #Z 1)) * (1 / (1 + (x ^2)))) by TAYLOR_1:1
.= (((arccot * ((id Z) ^)) . x) * (2 * x)) + ((x * (x #Z 1)) * (1 / (1 + (x ^2)))) by PREPOWER:35
.= (((arccot * ((id Z) ^)) . x) * (2 * x)) + ((x ^2) / (1 + (x ^2))) by PREPOWER:35
.= ((arccot . (((id Z) ^) . x)) * (2 * x)) + ((x ^2) / (1 + (x ^2))) by A7, A12, FUNCT_1:12
.= ((arccot . (((id Z) . x) ")) * (2 * x)) + ((x ^2) / (1 + (x ^2))) by A11, A12, RFUNCT_1:def_2
.= ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) by A12, FUNCT_1:18 ;
hence  ((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( g (#) (arccot * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
((g (#) (arccot * ((id Z) ^))) `| Z) . x = ((2 * x) * (arccot . (1 / x))) + ((x ^2) / (1 + (x ^2))) ) ) by A2, A8, A9, FDIFF_1:21; ::_thesis: verum
end;

theorem Th67: :: FDIFF_11:67
for Z being open Subset of REAL st Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
arctan . x <> 0 ) holds
( arctan ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan ^) `| Z) . x = - (1 / (((arctan . x) ^2) * (1 + (x ^2)))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
arctan . x <> 0 ) implies ( arctan ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan ^) `| Z) . x = - (1 / (((arctan . x) ^2) * (1 + (x ^2)))) ) ) )
assume that
A1: Z c= ].(- 1),1.[ and
A2: for x being Real st x in Z holds
arctan . x <> 0 ; ::_thesis: ( arctan ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan ^) `| Z) . x = - (1 / (((arctan . x) ^2) * (1 + (x ^2)))) ) )
A3: arctan is_differentiable_on Z by A1, SIN_COS9:81;
then A4: arctan ^ is_differentiable_on Z by A2, FDIFF_2:22;
 for x being Real st x in Z holds
((arctan ^) `| Z) . x = - (1 / (((arctan . x) ^2) * (1 + (x ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies ((arctan ^) `| Z) . x = - (1 / (((arctan . x) ^2) * (1 + (x ^2)))) )
assume A5: x in Z ; ::_thesis: ((arctan ^) `| Z) . x = - (1 / (((arctan . x) ^2) * (1 + (x ^2))))
then A6: ( arctan . x <> 0 & arctan is_differentiable_in x ) by A2, A3, FDIFF_1:9;
((arctan ^) `| Z) . x = diff ((arctan ^),x) by A4, A5, FDIFF_1:def_7
.= - ((diff (arctan,x)) / ((arctan . x) ^2)) by A6, FDIFF_2:15
.= - (((arctan `| Z) . x) / ((arctan . x) ^2)) by A3, A5, FDIFF_1:def_7
.= - ((1 / (1 + (x ^2))) / ((arctan . x) ^2)) by A1, A5, SIN_COS9:81
.= - (1 / (((arctan . x) ^2) * (1 + (x ^2)))) by XCMPLX_1:78 ;
hence  ((arctan ^) `| Z) . x = - (1 / (((arctan . x) ^2) * (1 + (x ^2)))) ; ::_thesis: verum
end;
hence  ( arctan ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan ^) `| Z) . x = - (1 / (((arctan . x) ^2) * (1 + (x ^2)))) ) ) by A2, A3, FDIFF_2:22; ::_thesis: verum
end;

theorem Th68: :: FDIFF_11:68
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 / (((arccot . x) ^2) * (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 / (((arccot . x) ^2) * (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 / (((arccot . x) ^2) * (1 + (x ^2))) ) )
then A2: arccot is_differentiable_on Z by SIN_COS9:82;
A3: for x being Real st x in Z holds
arccot . x <> 0
proof
 PI in ].0,4.[ by SIN_COS:def_28;
then  PI > 0 by XXREAL_1:4;
then A4: PI / 4 > 0 / 4 by XREAL_1:74;
let x be Real; ::_thesis: ( x in Z implies arccot . x <> 0 )
assume A5: x in Z ; ::_thesis: arccot . x <> 0
assume A6: arccot . x = 0 ; ::_thesis: contradiction
 ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  Z c= [.(- 1),1.] by A1, XBOOLE_1:1;
then  x in [.(- 1),1.] by A5;
then  0 in arccot .: [.(- 1),1.] by A6, FUNCT_1:def_6, SIN_COS9:24;
then  0 in [.(PI / 4),((3 / 4) * PI).] by RELAT_1:115, SIN_COS9:56;
hence  contradiction by A4, XXREAL_1:1; ::_thesis: verum
end;
then A7: arccot ^ is_differentiable_on Z by A2, FDIFF_2:22;
 for x being Real st x in Z holds
((arccot ^) `| Z) . x = 1 / (((arccot . x) ^2) * (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((arccot ^) `| Z) . x = 1 / (((arccot . x) ^2) * (1 + (x ^2))) )
assume A8: x in Z ; ::_thesis: ((arccot ^) `| Z) . x = 1 / (((arccot . x) ^2) * (1 + (x ^2)))
then A9: ( arccot . x <> 0 & arccot is_differentiable_in x ) by A3, A2, FDIFF_1:9;
((arccot ^) `| Z) . x = diff ((arccot ^),x) by A7, A8, FDIFF_1:def_7
.= - ((diff (arccot,x)) / ((arccot . x) ^2)) by A9, FDIFF_2:15
.= - (((arccot `| Z) . x) / ((arccot . x) ^2)) by A2, A8, FDIFF_1:def_7
.= - ((- (1 / (1 + (x ^2)))) / ((arccot . x) ^2)) by A1, A8, SIN_COS9:82
.= (1 / (1 + (x ^2))) / ((arccot . x) ^2)
.= 1 / (((arccot . x) ^2) * (1 + (x ^2))) by XCMPLX_1:78 ;
hence  ((arccot ^) `| Z) . x = 1 / (((arccot . x) ^2) * (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( arccot ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot ^) `| Z) . x = 1 / (((arccot . x) ^2) * (1 + (x ^2))) ) ) by A3, A2, FDIFF_2:22; ::_thesis: verum
end;

theorem :: FDIFF_11:69
for n being Element of NAT
for Z being open Subset of REAL st Z c= dom ((1 / n) (#) ((#Z n) * (arctan ^))) & Z c= ].(- 1),1.[ & n > 0 & ( for x being Real st x in Z holds
arctan . x <> 0 ) holds
( (1 / n) (#) ((#Z n) * (arctan ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((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 ((1 / n) (#) ((#Z n) * (arctan ^))) & Z c= ].(- 1),1.[ & n > 0 & ( for x being Real st x in Z holds
arctan . x <> 0 ) holds
( (1 / n) (#) ((#Z n) * (arctan ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2)))) ) )
let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((1 / n) (#) ((#Z n) * (arctan ^))) & Z c= ].(- 1),1.[ & n > 0 & ( for x being Real st x in Z holds
arctan . x <> 0 ) implies ( (1 / n) (#) ((#Z n) * (arctan ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2)))) ) ) )
assume that
A1: Z c= dom ((1 / n) (#) ((#Z n) * (arctan ^))) and
A2: Z c= ].(- 1),1.[ and
A3: n > 0 and
A4: for x being Real st x in Z holds
arctan . x <> 0 ; ::_thesis: ( (1 / n) (#) ((#Z n) * (arctan ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2)))) ) )
A5: Z c= dom ((#Z n) * (arctan ^)) by A1, VALUED_1:def_5;
A6: arctan ^ is_differentiable_on Z by A2, A4, Th67;
 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  x in Z ; ::_thesis: (#Z n) * (arctan ^) is_differentiable_in x
then  arctan ^ is_differentiable_in x by A6, FDIFF_1:9;
hence  (#Z n) * (arctan ^) is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum
end;
then A7: (#Z n) * (arctan ^) is_differentiable_on Z by A5, FDIFF_1:9;
 for y being set st y in Z holds
y in dom (arctan ^) by A5, FUNCT_1:11;
then A8: Z c= dom (arctan ^) by TARSKI:def_3;
 for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2))))
proof
let x be Real; ::_thesis: ( x in Z implies (((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2)))) )
assume A9: x in Z ; ::_thesis: (((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2))))
then A10: arctan ^ is_differentiable_in x by A6, FDIFF_1:9;
A11: (arctan ^) . x = 1 / (arctan . x) by A8, A9, RFUNCT_1:def_2;
(((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = (1 / n) * (diff (((#Z n) * (arctan ^)),x)) by A1, A7, A9, FDIFF_1:20
.= (1 / n) * ((n * (((arctan ^) . x) #Z (n - 1))) * (diff ((arctan ^),x))) by A10, TAYLOR_1:3
.= (1 / n) * ((n * (((arctan ^) . x) #Z (n - 1))) * (((arctan ^) `| Z) . x)) by A6, A9, FDIFF_1:def_7
.= (1 / n) * ((n * (((arctan ^) . x) #Z (n - 1))) * (- (1 / (((arctan . x) ^2) * (1 + (x ^2)))))) by A2, A4, A9, Th67
.= - ((((1 / n) * n) * (((arctan ^) . x) #Z (n - 1))) * (1 / (((arctan . x) ^2) * (1 + (x ^2)))))
.= - ((1 * (((arctan ^) . x) #Z (n - 1))) * (1 / (((arctan . x) ^2) * (1 + (x ^2))))) by A3, XCMPLX_1:106
.= - (((1 / (arctan . x)) #Z (n - 1)) * (1 / (((arctan . x) #Z 2) * (1 + (x ^2))))) by A11, FDIFF_7:1
.= - ((1 / ((arctan . x) #Z (n - 1))) / (((arctan . x) #Z 2) * (1 + (x ^2)))) by PREPOWER:42
.= - (1 / (((arctan . x) #Z (n - 1)) * (((arctan . x) #Z 2) * (1 + (x ^2))))) by XCMPLX_1:78
.= - (1 / ((((arctan . x) #Z (n - 1)) * ((arctan . x) #Z 2)) * (1 + (x ^2))))
.= - (1 / (((arctan . x) #Z ((n - 1) + 2)) * (1 + (x ^2)))) by A4, A9, PREPOWER:44
.= - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2)))) ;
hence  (((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2)))) ; ::_thesis: verum
end;
hence  ( (1 / n) (#) ((#Z n) * (arctan ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arctan ^))) `| Z) . x = - (1 / (((arctan . x) #Z (n + 1)) * (1 + (x ^2)))) ) ) by A1, A7, FDIFF_1:20; ::_thesis: verum
end;

theorem :: FDIFF_11:70
for n being Element of NAT
for Z being open Subset of REAL st Z c= dom ((1 / n) (#) ((#Z n) * (arccot ^))) & Z c= ].(- 1),1.[ & n > 0 holds
( (1 / n) (#) ((#Z n) * (arccot ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((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 ((1 / n) (#) ((#Z n) * (arccot ^))) & Z c= ].(- 1),1.[ & n > 0 holds
( (1 / n) (#) ((#Z n) * (arccot ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) ) )
let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((1 / n) (#) ((#Z n) * (arccot ^))) & Z c= ].(- 1),1.[ & n > 0 implies ( (1 / n) (#) ((#Z n) * (arccot ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom ((1 / n) (#) ((#Z n) * (arccot ^))) and
A2: Z c= ].(- 1),1.[ and
A3: n > 0 ; ::_thesis: ( (1 / n) (#) ((#Z n) * (arccot ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) ) )
A4: Z c= dom ((#Z n) * (arccot ^)) by A1, VALUED_1:def_5;
A5: for x being Real st x in Z holds
arccot . x <> 0
proof
 PI in ].0,4.[ by SIN_COS:def_28;
then  PI > 0 by XXREAL_1:4;
then A6: PI / 4 > 0 / 4 by XREAL_1:74;
let x be Real; ::_thesis: ( x in Z implies arccot . x <> 0 )
assume A7: x in Z ; ::_thesis: arccot . x <> 0
assume A8: arccot . x = 0 ; ::_thesis: contradiction
 ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then  Z c= [.(- 1),1.] by A2, XBOOLE_1:1;
then  x in [.(- 1),1.] by A7;
then  0 in arccot .: [.(- 1),1.] by A8, FUNCT_1:def_6, SIN_COS9:24;
then  0 in [.(PI / 4),((3 / 4) * PI).] by RELAT_1:115, SIN_COS9:56;
hence  contradiction by A6, XXREAL_1:1; ::_thesis: verum
end;
A9: arccot ^ is_differentiable_on Z by A2, Th68;
 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  x in Z ; ::_thesis: (#Z n) * (arccot ^) is_differentiable_in x
then  arccot ^ is_differentiable_in x by A9, FDIFF_1:9;
hence  (#Z n) * (arccot ^) is_differentiable_in x by TAYLOR_1:3; ::_thesis: verum
end;
then A10: (#Z n) * (arccot ^) is_differentiable_on Z by A4, FDIFF_1:9;
 for y being set st y in Z holds
y in dom (arccot ^) by A4, FUNCT_1:11;
then A11: Z c= dom (arccot ^) by TARSKI:def_3;
 for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies (((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) )
assume A12: x in Z ; ::_thesis: (((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2)))
then A13: arccot ^ is_differentiable_in x by A9, FDIFF_1:9;
A14: (arccot ^) . x = 1 / (arccot . x) by A11, A12, RFUNCT_1:def_2;
(((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = (1 / n) * (diff (((#Z n) * (arccot ^)),x)) by A1, A10, A12, FDIFF_1:20
.= (1 / n) * ((n * (((arccot ^) . x) #Z (n - 1))) * (diff ((arccot ^),x))) by A13, TAYLOR_1:3
.= (1 / n) * ((n * (((arccot ^) . x) #Z (n - 1))) * (((arccot ^) `| Z) . x)) by A9, A12, FDIFF_1:def_7
.= (1 / n) * ((n * (((arccot ^) . x) #Z (n - 1))) * (1 / (((arccot . x) ^2) * (1 + (x ^2))))) by A2, A12, Th68
.= (((1 / n) * n) * (((arccot ^) . x) #Z (n - 1))) * (1 / (((arccot . x) ^2) * (1 + (x ^2))))
.= (1 * (((arccot ^) . x) #Z (n - 1))) * (1 / (((arccot . x) ^2) * (1 + (x ^2)))) by A3, XCMPLX_1:106
.= ((1 / (arccot . x)) #Z (n - 1)) * (1 / (((arccot . x) #Z 2) * (1 + (x ^2)))) by A14, FDIFF_7:1
.= (1 / ((arccot . x) #Z (n - 1))) / (((arccot . x) #Z 2) * (1 + (x ^2))) by PREPOWER:42
.= 1 / (((arccot . x) #Z (n - 1)) * (((arccot . x) #Z 2) * (1 + (x ^2)))) by XCMPLX_1:78
.= 1 / ((((arccot . x) #Z (n - 1)) * ((arccot . x) #Z 2)) * (1 + (x ^2)))
.= 1 / (((arccot . x) #Z ((n - 1) + 2)) * (1 + (x ^2))) by A5, A12, PREPOWER:44
.= 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) ;
hence  (((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( (1 / n) (#) ((#Z n) * (arccot ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) ) ) by A1, A10, FDIFF_1:20; ::_thesis: verum
end;

theorem :: FDIFF_11:71
for Z being open Subset of REAL st Z c= dom (2 (#) ((#R (1 / 2)) * arctan)) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
arctan . x > 0 ) holds
( 2 (#) ((#R (1 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (2 (#) ((#R (1 / 2)) * arctan)) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
arctan . x > 0 ) implies ( 2 (#) ((#R (1 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ) ) )
assume that
A1: Z c= dom (2 (#) ((#R (1 / 2)) * arctan)) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
arctan . x > 0 ; ::_thesis: ( 2 (#) ((#R (1 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ) )
A4: for x being Real st x in Z holds
(#R (1 / 2)) * arctan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies (#R (1 / 2)) * arctan is_differentiable_in x )
assume A5: x in Z ; ::_thesis: (#R (1 / 2)) * arctan is_differentiable_in x
then A6: arctan . x > 0 by A3;
 arctan is_differentiable_on Z by A2, SIN_COS9:81;
then  arctan is_differentiable_in x by A5, FDIFF_1:9;
hence  (#R (1 / 2)) * arctan is_differentiable_in x by A6, TAYLOR_1:22; ::_thesis: verum
end;
 Z c= dom ((#R (1 / 2)) * arctan) by A1, VALUED_1:def_5;
then A7: (#R (1 / 2)) * arctan is_differentiable_on Z by A4, FDIFF_1:9;
 for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2))
proof
let x be Real; ::_thesis: ( x in Z implies ((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) )
assume A8: x in Z ; ::_thesis: ((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2))
then A9: arctan . x > 0 by A3;
A10: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A11: arctan is_differentiable_in x by A8, FDIFF_1:9;
((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = 2 * (diff (((#R (1 / 2)) * arctan),x)) by A1, A7, A8, FDIFF_1:20
.= 2 * (((1 / 2) * ((arctan . x) #R ((1 / 2) - 1))) * (diff (arctan,x))) by A11, A9, TAYLOR_1:22
.= 2 * (((1 / 2) * ((arctan . x) #R ((1 / 2) - 1))) * ((arctan `| Z) . x)) by A8, A10, FDIFF_1:def_7
.= 2 * (((1 / 2) * ((arctan . x) #R ((1 / 2) - 1))) * (1 / (1 + (x ^2)))) by A2, A8, SIN_COS9:81
.= ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ;
hence  ((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ; ::_thesis: verum
end;
hence  ( 2 (#) ((#R (1 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ) ) by A1, A7, FDIFF_1:20; ::_thesis: verum
end;

theorem :: FDIFF_11:72
for Z being open Subset of REAL st Z c= dom (2 (#) ((#R (1 / 2)) * arccot)) & Z c= ].(- 1),1.[ holds
( 2 (#) ((#R (1 / 2)) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom (2 (#) ((#R (1 / 2)) * arccot)) & Z c= ].(- 1),1.[ implies ( 2 (#) ((#R (1 / 2)) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (2 (#) ((#R (1 / 2)) * arccot)) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( 2 (#) ((#R (1 / 2)) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) ) )
A3: for x being Real st x in Z holds
arccot . x > 0
proof
let x be Real; ::_thesis: ( x in Z implies arccot . x > 0 )
 ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then A4: Z c= [.(- 1),1.] by A2, XBOOLE_1:1;
assume  x in Z ; ::_thesis: arccot . x > 0
then  x in [.(- 1),1.] by A4;
then  arccot . x in arccot .: [.(- 1),1.] by FUNCT_1:def_6, SIN_COS9:24;
then  arccot . x in [.(PI / 4),((3 / 4) * PI).] by RELAT_1:115, SIN_COS9:56;
then  arccot . x >= PI / 4 by XXREAL_1:1;
then A5: (arccot . x) + 0 > (PI / 4) + (- (PI / 4)) by XREAL_1:8;
assume  arccot . x <= 0 ; ::_thesis: contradiction
hence  contradiction by A5; ::_thesis: verum
end;
A6: for x being Real st x in Z holds
(#R (1 / 2)) * arccot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies (#R (1 / 2)) * arccot is_differentiable_in x )
assume A7: x in Z ; ::_thesis: (#R (1 / 2)) * arccot is_differentiable_in x
then A8: arccot . x > 0 by A3;
 arccot is_differentiable_on Z by A2, SIN_COS9:82;
then  arccot is_differentiable_in x by A7, FDIFF_1:9;
hence  (#R (1 / 2)) * arccot is_differentiable_in x by A8, TAYLOR_1:22; ::_thesis: verum
end;
 Z c= dom ((#R (1 / 2)) * arccot) by A1, VALUED_1:def_5;
then A9: (#R (1 / 2)) * arccot is_differentiable_on Z by A6, FDIFF_1:9;
 for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies ((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) )
assume A10: x in Z ; ::_thesis: ((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2)))
then A11: arccot . x > 0 by A3;
A12: arccot is_differentiable_on Z by A2, SIN_COS9:82;
then A13: arccot is_differentiable_in x by A10, FDIFF_1:9;
((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = 2 * (diff (((#R (1 / 2)) * arccot),x)) by A1, A9, A10, FDIFF_1:20
.= 2 * (((1 / 2) * ((arccot . x) #R ((1 / 2) - 1))) * (diff (arccot,x))) by A13, A11, TAYLOR_1:22
.= 2 * (((1 / 2) * ((arccot . x) #R ((1 / 2) - 1))) * ((arccot `| Z) . x)) by A10, A12, FDIFF_1:def_7
.= 2 * (((1 / 2) * ((arccot . x) #R ((1 / 2) - 1))) * (- (1 / (1 + (x ^2))))) by A2, A10, SIN_COS9:82
.= - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) ;
hence  ((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( 2 (#) ((#R (1 / 2)) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (- (1 / 2))) / (1 + (x ^2))) ) ) by A1, A9, FDIFF_1:20; ::_thesis: verum
end;

theorem :: FDIFF_11:73
for Z being open Subset of REAL st Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * arctan)) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
arctan . x > 0 ) holds
( (2 / 3) (#) ((#R (3 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (1 / 2)) / (1 + (x ^2)) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * arctan)) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
arctan . x > 0 ) implies ( (2 / 3) (#) ((#R (3 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (1 / 2)) / (1 + (x ^2)) ) ) )
assume that
A1: Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * arctan)) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
arctan . x > 0 ; ::_thesis: ( (2 / 3) (#) ((#R (3 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (1 / 2)) / (1 + (x ^2)) ) )
A4: for x being Real st x in Z holds
(#R (3 / 2)) * arctan is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies (#R (3 / 2)) * arctan is_differentiable_in x )
assume A5: x in Z ; ::_thesis: (#R (3 / 2)) * arctan is_differentiable_in x
then A6: arctan . x > 0 by A3;
 arctan is_differentiable_on Z by A2, SIN_COS9:81;
then  arctan is_differentiable_in x by A5, FDIFF_1:9;
hence  (#R (3 / 2)) * arctan is_differentiable_in x by A6, TAYLOR_1:22; ::_thesis: verum
end;
 Z c= dom ((#R (3 / 2)) * arctan) by A1, VALUED_1:def_5;
then A7: (#R (3 / 2)) * arctan is_differentiable_on Z by A4, FDIFF_1:9;
 for x being Real st x in Z holds
(((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (1 / 2)) / (1 + (x ^2))
proof
let x be Real; ::_thesis: ( x in Z implies (((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (1 / 2)) / (1 + (x ^2)) )
assume A8: x in Z ; ::_thesis: (((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (1 / 2)) / (1 + (x ^2))
then A9: arctan . x > 0 by A3;
A10: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A11: arctan is_differentiable_in x by A8, FDIFF_1:9;
(((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = (2 / 3) * (diff (((#R (3 / 2)) * arctan),x)) by A1, A7, A8, FDIFF_1:20
.= (2 / 3) * (((3 / 2) * ((arctan . x) #R ((3 / 2) - 1))) * (diff (arctan,x))) by A11, A9, TAYLOR_1:22
.= (2 / 3) * (((3 / 2) * ((arctan . x) #R ((3 / 2) - 1))) * ((arctan `| Z) . x)) by A8, A10, FDIFF_1:def_7
.= (2 / 3) * (((3 / 2) * ((arctan . x) #R ((3 / 2) - 1))) * (1 / (1 + (x ^2)))) by A2, A8, SIN_COS9:81
.= ((arctan . x) #R (1 / 2)) / (1 + (x ^2)) ;
hence  (((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (1 / 2)) / (1 + (x ^2)) ; ::_thesis: verum
end;
hence  ( (2 / 3) (#) ((#R (3 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / 3) (#) ((#R (3 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (1 / 2)) / (1 + (x ^2)) ) ) by A1, A7, FDIFF_1:20; ::_thesis: verum
end;

theorem :: FDIFF_11:74
for Z being open Subset of REAL st Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * arccot)) & Z c= ].(- 1),1.[ holds
( (2 / 3) (#) ((#R (3 / 2)) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) ) )
proof
let Z be open Subset of REAL; ::_thesis: ( Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * arccot)) & Z c= ].(- 1),1.[ implies ( (2 / 3) (#) ((#R (3 / 2)) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * arccot)) and
A2: Z c= ].(- 1),1.[ ; ::_thesis: ( (2 / 3) (#) ((#R (3 / 2)) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) ) )
A3: for x being Real st x in Z holds
arccot . x > 0
proof
let x be Real; ::_thesis: ( x in Z implies arccot . x > 0 )
 ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then A4: Z c= [.(- 1),1.] by A2, XBOOLE_1:1;
assume  x in Z ; ::_thesis: arccot . x > 0
then  x in [.(- 1),1.] by A4;
then  arccot . x in arccot .: [.(- 1),1.] by FUNCT_1:def_6, SIN_COS9:24;
then  arccot . x in [.(PI / 4),((3 / 4) * PI).] by RELAT_1:115, SIN_COS9:56;
then  arccot . x >= PI / 4 by XXREAL_1:1;
then A5: (arccot . x) + 0 > (PI / 4) + (- (PI / 4)) by XREAL_1:8;
assume  arccot . x <= 0 ; ::_thesis: contradiction
hence  contradiction by A5; ::_thesis: verum
end;
A6: for x being Real st x in Z holds
(#R (3 / 2)) * arccot is_differentiable_in x
proof
let x be Real; ::_thesis: ( x in Z implies (#R (3 / 2)) * arccot is_differentiable_in x )
assume A7: x in Z ; ::_thesis: (#R (3 / 2)) * arccot is_differentiable_in x
then A8: arccot . x > 0 by A3;
 arccot is_differentiable_on Z by A2, SIN_COS9:82;
then  arccot is_differentiable_in x by A7, FDIFF_1:9;
hence  (#R (3 / 2)) * arccot is_differentiable_in x by A8, TAYLOR_1:22; ::_thesis: verum
end;
 Z c= dom ((#R (3 / 2)) * arccot) by A1, VALUED_1:def_5;
then A9: (#R (3 / 2)) * arccot is_differentiable_on Z by A6, FDIFF_1:9;
 for x being Real st x in Z holds
(((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (1 / 2)) / (1 + (x ^2)))
proof
let x be Real; ::_thesis: ( x in Z implies (((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) )
assume A10: x in Z ; ::_thesis: (((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (1 / 2)) / (1 + (x ^2)))
then A11: arccot . x > 0 by A3;
A12: arccot is_differentiable_on Z by A2, SIN_COS9:82;
then A13: arccot is_differentiable_in x by A10, FDIFF_1:9;
(((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = (2 / 3) * (diff (((#R (3 / 2)) * arccot),x)) by A1, A9, A10, FDIFF_1:20
.= (2 / 3) * (((3 / 2) * ((arccot . x) #R ((3 / 2) - 1))) * (diff (arccot,x))) by A13, A11, TAYLOR_1:22
.= (2 / 3) * (((3 / 2) * ((arccot . x) #R ((3 / 2) - 1))) * ((arccot `| Z) . x)) by A10, A12, FDIFF_1:def_7
.= (2 / 3) * (((3 / 2) * ((arccot . x) #R ((3 / 2) - 1))) * (- (1 / (1 + (x ^2))))) by A2, A10, SIN_COS9:82
.= - (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) ;
hence  (((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) ; ::_thesis: verum
end;
hence  ( (2 / 3) (#) ((#R (3 / 2)) * arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / 3) (#) ((#R (3 / 2)) * arccot)) `| Z) . x = - (((arccot . x) #R (1 / 2)) / (1 + (x ^2))) ) ) by A1, A9, FDIFF_1:20; ::_thesis: verum
end;