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_1:16;
sin is_differentiable_in (arctan - arccot ) . x by SIN_COS:69;
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_1:16;
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:69;
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_1:16;
((sin * (arctan - arccot )) `| Z) . x = diff (sin * (arctan - arccot )),x by A9, A11, FDIFF_1:def 8
.= (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:69
.= (cos . ((arctan - arccot ) . x)) * (((arctan - arccot ) `| Z) . x) by A6, A11, FDIFF_1:def 8
.= (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_1:16; :: thesis: verum