let Z be open Subset of REAL ; :: thesis: ( Z c= dom (cot * cot ) implies ( cot * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * cot ) `| Z) . x = (1 / ((sin . (cot . x)) ^2 )) * (1 / ((sin . x) ^2 )) ) ) )
assume A1: Z c= dom (cot * cot ) ; :: thesis: ( cot * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * cot ) `| Z) . x = (1 / ((sin . (cot . x)) ^2 )) * (1 / ((sin . x) ^2 )) ) )
A2: for x being Real st x in Z holds
sin . x <> 0

proof

let x be Real; :: thesis: ( x in Z implies sin . x <> 0 )
assume x in Z ; :: thesis: sin . x <> 0
then x in dom (cos / sin ) by A1, FUNCT_1:21;
hence sin . x <> 0 by FDIFF_8:2; :: thesis: verum

end;

A3: for x being Real st x in Z holds
sin . (cot . x) <> 0

proof

let x be Real; :: thesis: ( x in Z implies sin . (cot . x) <> 0 )
assume x in Z ; :: thesis: sin . (cot . x) <> 0
then cot . x in dom cot by A1, FUNCT_1:21;
hence sin . (cot . x) <> 0 by FDIFF_8:2; :: thesis: verum

end;

A4: for x being Real st x in Z holds
cot * cot is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cot * cot is_differentiable_in x )
assume A5: x in Z ; :: thesis: cot * cot is_differentiable_in x
then sin . x <> 0 by A2;
then A6: cot is_differentiable_in x by FDIFF_7:47;
sin . (cot . x) <> 0 by A3, A5;
then cot is_differentiable_in cot . x by FDIFF_7:47;
hence cot * cot is_differentiable_in x by A6, FDIFF_2:13; :: thesis: verum

end;

then A7: cot * cot is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((cot * cot ) `| Z) . x = (1 / ((sin . (cot . x)) ^2 )) * (1 / ((sin . x) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((cot * cot ) `| Z) . x = (1 / ((sin . (cot . x)) ^2 )) * (1 / ((sin . x) ^2 )) )
assume A8: x in Z ; :: thesis: ((cot * cot ) `| Z) . x = (1 / ((sin . (cot . x)) ^2 )) * (1 / ((sin . x) ^2 ))
then A9: sin . x <> 0 by A2;
then A10: cot is_differentiable_in x by FDIFF_7:47;
A11: sin . (cot . x) <> 0 by A3, A8;
then cot is_differentiable_in cot . x by FDIFF_7:47;
then diff (cot * cot ),x = (diff cot ,(cot . x)) * (diff cot ,x) by A10, FDIFF_2:13
.= (- (1 / ((sin . (cot . x)) ^2 ))) * (diff cot ,x) by A11, FDIFF_7:47
.= (- (1 / ((sin . (cot . x)) ^2 ))) * (- (1 / ((sin . x) ^2 ))) by A9, FDIFF_7:47 ;
hence ((cot * cot ) `| Z) . x = (1 / ((sin . (cot . x)) ^2 )) * (1 / ((sin . x) ^2 )) by A7, A8, FDIFF_1:def 8; :: thesis: verum

end;

hence ( cot * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * cot ) `| Z) . x = (1 / ((sin . (cot . x)) ^2 )) * (1 / ((sin . x) ^2 )) ) ) by A1, A4, FDIFF_1:16; :: thesis: verum