let Z be open Subset of REAL; :: thesis: ( Z c= dom (sec * cot) implies ( sec * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * cot) `| Z) . x = - (((sin . (cot . x)) / ((sin . x) ^2)) / ((cos . (cot . x)) ^2)) ) ) )
assume A1: Z c= dom (sec * cot) ; :: thesis: ( sec * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * cot) `| Z) . x = - (((sin . (cot . x)) / ((sin . x) ^2)) / ((cos . (cot . x)) ^2)) ) )
A2: for x being Real st x in Z holds
cos . (cot . x) <> 0

proof

let x be Real; :: thesis: ( x in Z implies cos . (cot . x) <> 0 )
assume x in Z ; :: thesis: cos . (cot . x) <> 0
then cot . x in dom sec by A1, FUNCT_1:11;
hence cos . (cot . x) <> 0 by RFUNCT_1:3; :: thesis: verum

end;

dom (sec * 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
sec * cot is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies sec * cot is_differentiable_in x )
assume A5: x in Z ; :: thesis: sec * 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;
cos . (cot . x) <> 0 by A2, A5;
then sec is_differentiable_in cot . x by Th1;
hence sec * cot is_differentiable_in x by A6, FDIFF_2:13; :: thesis: verum

end;

then A7: sec * cot is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((sec * cot) `| Z) . x = - (((sin . (cot . x)) / ((sin . x) ^2)) / ((cos . (cot . x)) ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((sec * cot) `| Z) . x = - (((sin . (cot . x)) / ((sin . x) ^2)) / ((cos . (cot . x)) ^2)) )
assume A8: x in Z ; :: thesis: ((sec * cot) `| Z) . x = - (((sin . (cot . x)) / ((sin . x) ^2)) / ((cos . (cot . x)) ^2))
then A9: sin . x <> 0 by A3, FDIFF_8:2;
then A10: cot is_differentiable_in x by FDIFF_7:47;
A11: cos . (cot . x) <> 0 by A2, A8;
then sec is_differentiable_in cot . x by Th1;
then diff ((sec * cot),x) = (diff (sec,(cot . x))) * (diff (cot,x)) by A10, FDIFF_2:13
.= ((sin . (cot . x)) / ((cos . (cot . x)) ^2)) * (diff (cot,x)) by A11, Th1
.= (- (1 / ((sin . x) ^2))) * ((sin . (cot . x)) / ((cos . (cot . x)) ^2)) by A9, FDIFF_7:47
.= - (((sin . (cot . x)) / ((sin . x) ^2)) / ((cos . (cot . x)) ^2)) ;
hence ((sec * cot) `| Z) . x = - (((sin . (cot . x)) / ((sin . x) ^2)) / ((cos . (cot . x)) ^2)) by A7, A8, FDIFF_1:def 7; :: thesis: verum

end;

hence ( sec * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * cot) `| Z) . x = - (((sin . (cot . x)) / ((sin . x) ^2)) / ((cos . (cot . x)) ^2)) ) ) by A1, A4, FDIFF_1:9; :: thesis: verum