let Z be open Subset of REAL ; :: thesis:
assume that
A1: Z c= dom (tan - sec ) and
A2: for x being Real st x in Z holds
( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) ; :: thesis:
Z c= (dom tan ) /\ (dom (cos ^ )) by A1, VALUED_1:12;
then A3: ( Z c= dom tan & Z c= dom (cos ^ ) ) by XBOOLE_1:18;
then A4: for x being Real st x in Z holds
cos . x <> 0 by FDIFF_8:1;
then A5: cos ^ is_differentiable_on Z by FDIFF_4:39;
for x being Real st x in Z holds
tan is_differentiable_in x

let x be Real; :: thesis:
assume x in Z ; :: thesis:
then cos . x <> 0 by A3, FDIFF_8:1;
hence tan is_differentiable_in x by FDIFF_7:46; :: thesis:

end;

then A6: tan is_differentiable_on Z by A3, FDIFF_1:16;
for x being Real st x in Z holds
((tan - sec ) `| Z) . x = 1 / (1 + (sin . x))

let x be Real; :: thesis:
assume A7: x in Z ; :: thesis:
then A8: ( cos . x <> 0 & 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) by A2, A3, FDIFF_8:1;
((tan - sec ) `| Z) . x = (diff tan ,x) - (diff (cos ^ ),x) by A1, A5, A6, A7, FDIFF_1:27
.= (1 / ((cos . x) ^2 )) - (diff (cos ^ ),x) by A8, FDIFF_7:46
.= (1 / ((cos . x) ^2 )) - (((cos ^ ) `| Z) . x) by A5, A7, FDIFF_1:def 8
.= (1 / ((cos . x) ^2 )) - ((sin . x) / ((cos . x) ^2 )) by A4, A7, FDIFF_4:39
.= (1 - (sin . x)) / ((((cos . x) ^2 ) + ((sin . x) ^2 )) - ((sin . x) ^2 )) by XCMPLX_1:121
.= (1 - (sin . x)) / (1 - ((sin . x) ^2 )) by SIN_COS:31
.= (1 - (sin . x)) / ((1 + (sin . x)) * (1 - (sin . x)))
.= ((1 - (sin . x)) / (1 - (sin . x))) / (1 + (sin . x)) by XCMPLX_1:79
.= 1 / (1 + (sin . x)) by A8, XCMPLX_1:60 ;
hence ((tan - sec ) `| Z) . x = 1 / (1 + (sin . x)) ; :: thesis:

end;

hence ( tan - sec is_differentiable_on Z & ( for x being Real st x in Z holds
((tan - sec ) `| Z) . x = 1 / (1 + (sin . x)) ) ) by A1, A5, A6, FDIFF_1:27; :: thesis: