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:63;
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_1:9;
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:63;
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 7
.= - ((diff (cos,x)) / (1 + ((cos . x) ^2))) by A6, A8, SIN_COS9:86
.= - ((- (sin . x)) / (1 + ((cos . x) ^2))) by SIN_COS:63
.= (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_1:9; :: thesis: verum