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: ].(- 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;
then Z c= (dom sec ) /\ (dom (arctan - arccot )) by A1, XBOOLE_1:19;
then A6: Z c= dom (sec (#) (arctan - arccot )) by VALUED_1:def 4;
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_1:13;
hence sec is_differentiable_in x by FDIFF_9:1; :: thesis: verum

end;

then A7: sec is_differentiable_on Z by A1, FDIFF_1:16;
A8: ( 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;
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_1:13;
((sec (#) (arctan - arccot )) `| Z) . x = (((arctan - arccot ) . x) * (diff sec ,x)) + ((sec . x) * (diff (arctan - arccot ),x)) by A6, A7, A8, A9, FDIFF_1:29
.= (((arctan . x) - (arccot . x)) * (diff sec ,x)) + ((sec . x) * (diff (arctan - arccot ),x)) by A5, 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 A8, A9, FDIFF_1:def 8
.= ((((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 A6, A7, A8, FDIFF_1:29; :: thesis: verum