let Z be open Subset of REAL ; :: thesis: ( Z c= dom (cot * arccot ) & Z c= ].(- 1),1.[ implies ( cot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * arccot ) `| Z) . x = 1 / (((sin . (arccot . x)) ^2 ) * (1 + (x ^2 ))) ) ) )
assume that
A1: Z c= dom (cot * arccot ) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( cot * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * arccot ) `| Z) . x = 1 / (((sin . (arccot . x)) ^2 ) * (1 + (x ^2 ))) ) )
AA: for x being Real st x in Z holds
cot * arccot is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cot * arccot is_differentiable_in x )
assume A3: x in Z ; :: thesis: cot * arccot is_differentiable_in x
arccot is_differentiable_on Z by A2, SIN_COS9:82;
then A4: arccot is_differentiable_in x by A3, FDIFF_1:16;
arccot . x in dom cot by A1, A3, FUNCT_1:21;
then sin . (arccot . x) <> 0 by FDIFF_8:2;
then cot is_differentiable_in arccot . x by FDIFF_7:47;
hence cot * arccot is_differentiable_in x by A4, FDIFF_2:13; :: thesis: verum

end;

then A5: cot * arccot is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((cot * arccot ) `| Z) . x = 1 / (((sin . (arccot . x)) ^2 ) * (1 + (x ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies ((cot * arccot ) `| Z) . x = 1 / (((sin . (arccot . x)) ^2 ) * (1 + (x ^2 ))) )
assume A6: x in Z ; :: thesis: ((cot * arccot ) `| Z) . x = 1 / (((sin . (arccot . x)) ^2 ) * (1 + (x ^2 )))
A7: arccot is_differentiable_on Z by A2, SIN_COS9:82;
then A8: arccot is_differentiable_in x by A6, FDIFF_1:16;
arccot . x in dom cot by A1, A6, FUNCT_1:21;
then A9: sin . (arccot . x) <> 0 by FDIFF_8:2;
then A10: cot is_differentiable_in arccot . x by FDIFF_7:47;
((cot * arccot ) `| Z) . x = diff (cot * arccot ),x by A5, A6, FDIFF_1:def 8
.= (diff cot ,(arccot . x)) * (diff arccot ,x) by A8, A10, FDIFF_2:13
.= (- (1 / ((sin . (arccot . x)) ^2 ))) * (diff arccot ,x) by A9, FDIFF_7:47
.= - ((1 / ((sin . (arccot . x)) ^2 )) * (diff arccot ,x))
.= - ((1 / ((sin . (arccot . x)) ^2 )) * ((arccot `| Z) . x)) by A6, A7, FDIFF_1:def 8
.= - ((1 / ((sin . (arccot . x)) ^2 )) * (- (1 / (1 + (x ^2 ))))) by A2, A6, SIN_COS9:82
.= (1 / ((sin . (arccot . x)) ^2 )) * (1 / (1 + (x ^2 )))
.= 1 / (((sin . (arccot . x)) ^2 ) * (1 + (x ^2 ))) by XCMPLX_1:103 ;
hence ((cot * arccot ) `| Z) . x = 1 / (((sin . (arccot . x)) ^2 ) * (1 + (x ^2 ))) ; :: thesis: verum

end;

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