let Z be open Subset of REAL ; :: thesis: ( Z c= dom (arccot * cos ) & ( for x being Real st x in Z holds
( cos . x > - 1 & cos . x < 1 ) ) implies ( arccot * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * cos ) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2 )) ) ) )
assume that
A1: Z c= dom (arccot * cos ) and
A2: for x being Real st x in Z holds
( cos . x > - 1 & cos . x < 1 ) ; :: thesis: ( arccot * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * cos ) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2 )) ) )
A3: for x being Real st x in Z holds
arccot * cos is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies arccot * cos is_differentiable_in x )
assume x in Z ; :: thesis: arccot * cos is_differentiable_in x
then A4: ( cos . x > - 1 & cos . x < 1 ) by A2;
cos is_differentiable_in x by SIN_COS:68;
hence arccot * cos is_differentiable_in x by A4, SIN_COS9:86; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies ((arccot * cos ) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2 )) )
A6: cos is_differentiable_in x by SIN_COS:68;
assume A7: x in Z ; :: thesis: ((arccot * cos ) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2 ))
then A8: ( cos . x > - 1 & cos . x < 1 ) by A2;
((arccot * cos ) `| Z) . x = diff (arccot * cos ),x by A5, A7, FDIFF_1:def 8
.= - ((diff cos ,x) / (1 + ((cos . x) ^2 ))) by A6, A8, SIN_COS9:86
.= - ((- (sin . x)) / (1 + ((cos . x) ^2 ))) by SIN_COS:68
.= (sin . x) / (1 + ((cos . x) ^2 )) ;
hence ((arccot * cos ) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2 )) ; :: thesis: verum

end;

hence ( arccot * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * cos ) `| Z) . x = (sin . x) / (1 + ((cos . x) ^2 )) ) ) by A1, A3, FDIFF_1:16; :: thesis: verum