let Z be open Subset of REAL; :: thesis: ( Z c= dom (cosec * cot) implies ( cosec * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * cot) `| Z) . x = ((cos . (cot . x)) / ((sin . x) ^2)) / ((sin . (cot . x)) ^2) ) ) )
assume A1: Z c= dom (cosec * cot) ; :: thesis: ( cosec * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * cot) `| Z) . x = ((cos . (cot . x)) / ((sin . x) ^2)) / ((sin . (cot . x)) ^2) ) )
A2: 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 cosec by A1, FUNCT_1:11;
hence sin . (cot . x) <> 0 by RFUNCT_1:3; :: thesis: verum

end;

dom (cosec * cot) c= dom cot by RELAT_1:25;
then A3: Z c= dom cot by A1, XBOOLE_1:1;
A4: for x being Real st x in Z holds
cosec * cot is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cosec * cot is_differentiable_in x )
assume A5: x in Z ; :: thesis: cosec * cot is_differentiable_in x
then sin . x <> 0 by A3, FDIFF_8:2;
then A6: cot is_differentiable_in x by FDIFF_7:47;
sin . (cot . x) <> 0 by A2, A5;
then cosec is_differentiable_in cot . x by Th2;
hence cosec * cot is_differentiable_in x by A6, FDIFF_2:13; :: thesis: verum

end;

then A7: cosec * cot is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((cosec * cot) `| Z) . x = ((cos . (cot . x)) / ((sin . x) ^2)) / ((sin . (cot . x)) ^2)

proof

let x be Real; :: thesis: ( x in Z implies ((cosec * cot) `| Z) . x = ((cos . (cot . x)) / ((sin . x) ^2)) / ((sin . (cot . x)) ^2) )
assume A8: x in Z ; :: thesis: ((cosec * cot) `| Z) . x = ((cos . (cot . x)) / ((sin . x) ^2)) / ((sin . (cot . x)) ^2)
then A9: sin . x <> 0 by A3, FDIFF_8:2;
then A10: cot is_differentiable_in x by FDIFF_7:47;
A11: sin . (cot . x) <> 0 by A2, A8;
then cosec is_differentiable_in cot . x by Th2;
then diff ((cosec * cot),x) = (diff (cosec,(cot . x))) * (diff (cot,x)) by A10, FDIFF_2:13
.= (- ((cos . (cot . x)) / ((sin . (cot . x)) ^2))) * (diff (cot,x)) by A11, Th2
.= (- (1 / ((sin . x) ^2))) * (- ((cos . (cot . x)) / ((sin . (cot . x)) ^2))) by A9, FDIFF_7:47
.= ((cos . (cot . x)) / ((sin . x) ^2)) / ((sin . (cot . x)) ^2) ;
hence ((cosec * cot) `| Z) . x = ((cos . (cot . x)) / ((sin . x) ^2)) / ((sin . (cot . x)) ^2) by A7, A8, FDIFF_1:def 7; :: thesis: verum

end;

hence ( cosec * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * cot) `| Z) . x = ((cos . (cot . x)) / ((sin . x) ^2)) / ((sin . (cot . x)) ^2) ) ) by A1, A4, FDIFF_1:9; :: thesis: verum