let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL , REAL st Z c= dom ((arctan * (f ^ )) (#) (arccot * (f ^ ))) & ( for x being Real st x in Z holds
( f . x = x & (f ^ ) . x > - 1 & (f ^ ) . x < 1 ) ) holds
( (arctan * (f ^ )) (#) (arccot * (f ^ )) is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan * (f ^ )) (#) (arccot * (f ^ ))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2 )) ) )
let f be PartFunc of REAL , REAL ; :: thesis: ( Z c= dom ((arctan * (f ^ )) (#) (arccot * (f ^ ))) & ( for x being Real st x in Z holds
( f . x = x & (f ^ ) . x > - 1 & (f ^ ) . x < 1 ) ) implies ( (arctan * (f ^ )) (#) (arccot * (f ^ )) is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan * (f ^ )) (#) (arccot * (f ^ ))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2 )) ) ) )
assume that
A1: Z c= dom ((arctan * (f ^ )) (#) (arccot * (f ^ ))) and
A2: for x being Real st x in Z holds
( f . x = x & (f ^ ) . x > - 1 & (f ^ ) . x < 1 ) ; :: thesis: ( (arctan * (f ^ )) (#) (arccot * (f ^ )) is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan * (f ^ )) (#) (arccot * (f ^ ))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2 )) ) )
Z c= (dom (arctan * (f ^ ))) /\ (dom (arccot * (f ^ ))) by A1, VALUED_1:def 4;
then A3: ( Z c= dom (arctan * (f ^ )) & Z c= dom (arccot * (f ^ )) ) by XBOOLE_1:18;
then for y being set st y in Z holds
y in dom (f ^ ) by FUNCT_1:21;
then A4: Z c= dom (f ^ ) by TARSKI:def 3;
A5: ( arctan * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * (f ^ )) `| Z) . x = - (1 / (1 + (x ^2 ))) ) ) by A2, A3, SIN_COS9:111;
A6: ( arccot * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * (f ^ )) `| Z) . x = 1 / (1 + (x ^2 )) ) ) by A2, A3, SIN_COS9:112;
for x being Real st x in Z holds
(((arctan * (f ^ )) (#) (arccot * (f ^ ))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies (((arctan * (f ^ )) (#) (arccot * (f ^ ))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2 )) )
assume A8: x in Z ; :: thesis: (((arctan * (f ^ )) (#) (arccot * (f ^ ))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2 ))
(((arctan * (f ^ )) (#) (arccot * (f ^ ))) `| Z) . x = (((arccot * (f ^ )) . x) * (diff (arctan * (f ^ )),x)) + (((arctan * (f ^ )) . x) * (diff (arccot * (f ^ )),x)) by A1, A5, A6, A8, FDIFF_1:29
.= (((arccot * (f ^ )) . x) * (((arctan * (f ^ )) `| Z) . x)) + (((arctan * (f ^ )) . x) * (diff (arccot * (f ^ )),x)) by A5, A8, FDIFF_1:def 8
.= (((arccot * (f ^ )) . x) * (- (1 / (1 + (x ^2 ))))) + (((arctan * (f ^ )) . x) * (diff (arccot * (f ^ )),x)) by A2, A3, A8, SIN_COS9:111
.= (((arccot * (f ^ )) . x) * (- (1 / (1 + (x ^2 ))))) + (((arctan * (f ^ )) . x) * (((arccot * (f ^ )) `| Z) . x)) by A6, A8, FDIFF_1:def 8
.= (((arccot * (f ^ )) . x) * (- (1 / (1 + (x ^2 ))))) + (((arctan * (f ^ )) . x) * (1 / (1 + (x ^2 )))) by A2, A3, A8, SIN_COS9:112
.= ((arccot . ((f ^ ) . x)) * (- (1 / (1 + (x ^2 ))))) + (((arctan * (f ^ )) . x) * (1 / (1 + (x ^2 )))) by A3, A8, FUNCT_1:22
.= ((arccot . ((f . x) " )) * (- (1 / (1 + (x ^2 ))))) + (((arctan * (f ^ )) . x) * (1 / (1 + (x ^2 )))) by A4, A8, RFUNCT_1:def 8
.= ((arccot . (1 / x)) * (- (1 / (1 + (x ^2 ))))) + (((arctan * (f ^ )) . x) * (1 / (1 + (x ^2 )))) by A2, A8
.= ((arccot . (1 / x)) * (- (1 / (1 + (x ^2 ))))) + ((arctan . ((f ^ ) . x)) * (1 / (1 + (x ^2 )))) by A3, A8, FUNCT_1:22
.= ((arccot . (1 / x)) * (- (1 / (1 + (x ^2 ))))) + ((arctan . ((f . x) " )) * (1 / (1 + (x ^2 )))) by A4, A8, RFUNCT_1:def 8
.= (- ((arccot . (1 / x)) * (1 / (1 + (x ^2 ))))) + ((arctan . (1 / x)) * (1 / (1 + (x ^2 )))) by A2, A8
.= ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2 )) ;
hence (((arctan * (f ^ )) (#) (arccot * (f ^ ))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2 )) ; :: thesis: verum

end;

hence ( (arctan * (f ^ )) (#) (arccot * (f ^ )) is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan * (f ^ )) (#) (arccot * (f ^ ))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2 )) ) ) by A1, A5, A6, FDIFF_1:29; :: thesis: verum