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 arctan by SIN_COS9:23, XBOOLE_1:1;
then A4: Z c= dom arctan by A2, XBOOLE_1:1;
].(- 1),1.[ c= dom arccot by A3, SIN_COS9:24, XBOOLE_1:1;
then Z c= dom arccot 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 & ( for x being Real st x in Z holds
((arctan - arccot ) `| Z) . x = 2 / (1 + (x ^2 )) ) ) 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 A8: x in Z ; :: thesis: cos * (arctan - arccot ) is_differentiable_in x
A9: arctan - arccot is_differentiable_in x by A6, A8, FDIFF_1:16;
cos is_differentiable_in (arctan - arccot ) . x by SIN_COS:68;
hence cos * (arctan - arccot ) is_differentiable_in x by A9, FDIFF_2:13; :: thesis: verum

end;

then A10: cos * (arctan - arccot ) is_differentiable_on Z by A1, FDIFF_1:16;
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 ))) )
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_1:16;
A13: cos is_differentiable_in (arctan - arccot ) . x by SIN_COS:68;
((cos * (arctan - arccot )) `| Z) . x = diff (cos * (arctan - arccot )),x by A10, A11, FDIFF_1:def 8
.= (diff cos ,((arctan - arccot ) . x)) * (diff (arctan - arccot ),x) by A12, A13, FDIFF_2:13
.= (- (sin . ((arctan - arccot ) . x))) * (diff (arctan - arccot ),x) by SIN_COS:68
.= (- (sin . ((arctan - arccot ) . x))) * (((arctan - arccot ) `| Z) . x) by A6, A11, FDIFF_1:def 8
.= (- (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_1:16; :: thesis: verum