let Z be open Subset of REAL ; :: thesis: ( Z c= dom (cosec * tan ) implies ( - (cosec * tan ) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cosec * tan )) `| Z) . x = ((cos . (tan . x)) / ((cos . x) ^2 )) / ((sin . (tan . x)) ^2 ) ) ) )
assume A1: Z c= dom (cosec * tan ) ; :: thesis: ( - (cosec * tan ) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cosec * tan )) `| Z) . x = ((cos . (tan . x)) / ((cos . x) ^2 )) / ((sin . (tan . x)) ^2 ) ) )
then A2: Z c= dom (- (cosec * tan )) by VALUED_1:8;
A3: cosec * tan is_differentiable_on Z by A1, FDIFF_9:40;
dom (cosec * tan ) c= dom tan by RELAT_1:44;
then AA: Z c= dom tan by A1, XBOOLE_1:1;
A4: (- 1) (#) (cosec * tan ) is_differentiable_on Z by A2, A3, FDIFF_1:28, A;
A5: for x being Real st x in Z holds
sin . (tan . x) <> 0

proof

let x be Real; :: thesis: ( x in Z implies sin . (tan . x) <> 0 )
assume x in Z ; :: thesis: sin . (tan . x) <> 0
then tan . x in dom cosec by A1, FUNCT_1:21;
hence sin . (tan . x) <> 0 by RFUNCT_1:13; :: thesis: verum

end;

for x being Real st x in Z holds
((- (cosec * tan )) `| Z) . x = ((cos . (tan . x)) / ((cos . x) ^2 )) / ((sin . (tan . x)) ^2 )

proof

let x be Real; :: thesis: ( x in Z implies ((- (cosec * tan )) `| Z) . x = ((cos . (tan . x)) / ((cos . x) ^2 )) / ((sin . (tan . x)) ^2 ) )
assume A6: x in Z ; :: thesis: ((- (cosec * tan )) `| Z) . x = ((cos . (tan . x)) / ((cos . x) ^2 )) / ((sin . (tan . x)) ^2 )
then A7: cos . x <> 0 by AA, FDIFF_8:1;
then A8: tan is_differentiable_in x by FDIFF_7:46;
A9: sin . (tan . x) <> 0 by A5, A6;
then A10: cosec is_differentiable_in tan . x by FDIFF_9:2;
A11: cosec * tan is_differentiable_in x by A3, A6, FDIFF_1:16;
((- (cosec * tan )) `| Z) . x = diff (- (cosec * tan )),x by A4, A6, FDIFF_1:def 8
.= (- 1) * (diff (cosec * tan ),x) by A11, FDIFF_1:23, A
.= (- 1) * ((diff cosec ,(tan . x)) * (diff tan ,x)) by A8, A10, FDIFF_2:13
.= (- 1) * ((- ((cos . (tan . x)) / ((sin . (tan . x)) ^2 ))) * (diff tan ,x)) by A9, FDIFF_9:2
.= (- 1) * ((1 / ((cos . x) ^2 )) * (- ((cos . (tan . x)) / ((sin . (tan . x)) ^2 )))) by A7, FDIFF_7:46
.= ((cos . (tan . x)) / ((cos . x) ^2 )) / ((sin . (tan . x)) ^2 ) ;
hence ((- (cosec * tan )) `| Z) . x = ((cos . (tan . x)) / ((cos . x) ^2 )) / ((sin . (tan . x)) ^2 ) ; :: thesis: verum

end;

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