let Z be open Subset of REAL; :: thesis: ( Z c= dom () implies ( sin (#) cot is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = ((cos . x) * (cot . x)) - (1 / (sin . x)) ) ) )

A1: for x being Real st x in Z holds
sin is_differentiable_in x by SIN_COS:64;
assume A2: Z c= dom () ; :: thesis: ( sin (#) cot is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = ((cos . x) * (cot . x)) - (1 / (sin . x)) ) )

then A3: Z c= () /\ () by VALUED_1:def 4;
then A4: Z c= dom cot by XBOOLE_1:18;
for x being Real st x in Z holds
cot is_differentiable_in x
proof end;
then A5: cot is_differentiable_on Z by ;
Z c= dom sin by ;
then A6: sin is_differentiable_on Z by ;
A7: for x being Real st x in Z holds
diff (cot,x) = - (1 / ((sin . x) ^2))
proof
let x be Real; :: thesis: ( x in Z implies diff (cot,x) = - (1 / ((sin . x) ^2)) )
assume x in Z ; :: thesis: diff (cot,x) = - (1 / ((sin . x) ^2))
then sin . x <> 0 by ;
hence diff (cot,x) = - (1 / ((sin . x) ^2)) by FDIFF_7:47; :: thesis: verum
end;
for x being Real st x in Z holds
(() `| Z) . x = ((cos . x) * (cot . x)) - (1 / (sin . x))
proof
let x be Real; :: thesis: ( x in Z implies (() `| Z) . x = ((cos . x) * (cot . x)) - (1 / (sin . x)) )
assume A8: x in Z ; :: thesis: (() `| Z) . x = ((cos . x) * (cot . x)) - (1 / (sin . x))
then (() `| Z) . x = ((diff (sin,x)) * (cot . x)) + ((sin . x) * (diff (cot,x))) by
.= ((cos . x) * (cot . x)) + ((sin . x) * (diff (cot,x))) by SIN_COS:64
.= ((cos . x) * (cot . x)) + ((sin . x) * (- (1 / ((sin . x) ^2)))) by A7, A8
.= ((cos . x) * (cot . x)) - ((sin . x) / ((sin . x) ^2))
.= ((cos . x) * (cot . x)) - (((sin . x) / (sin . x)) / (sin . x)) by XCMPLX_1:78
.= ((cos . x) * (cot . x)) - (1 / (sin . x)) by ;
hence ((sin (#) cot) `| Z) . x = ((cos . x) * (cot . x)) - (1 / (sin . x)) ; :: thesis: verum
end;
hence ( sin (#) cot is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = ((cos . x) * (cot . x)) - (1 / (sin . x)) ) ) by ; :: thesis: verum