let Z be open Subset of REAL ; :: thesis: ( Z c= dom (ln * sec ) implies ( ln * sec is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * sec ) `| Z) . x = (sin . x) / (cos . x) ) ) )
assume A1: Z c= dom (ln * sec ) ; :: thesis: ( ln * sec is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * sec ) `| Z) . x = (sin . x) / (cos . x) ) )
A2: for x being Real st x in Z holds
cos . x <> 0

proof

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

end;

A3: for x being Real st x in Z holds
sec . x > 0

proof

let x be Real; :: thesis: ( x in Z implies sec . x > 0 )
assume x in Z ; :: thesis: sec . x > 0
then sec . x in right_open_halfline 0 by A1, FUNCT_1:21, TAYLOR_1:18;
then ex g being Real st
( sec . x = g & 0 < g ) by Lm1;
hence sec . x > 0 ; :: thesis: verum

end;

dom (ln * sec ) c= dom sec by RELAT_1:44;
then A4: Z c= dom sec by A1, XBOOLE_1:1;
A5: for x being Real st x in Z holds
sec is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies sec is_differentiable_in x )
assume x in Z ; :: thesis: sec is_differentiable_in x
then cos . x <> 0 by A2;
hence sec is_differentiable_in x by Th1; :: thesis: verum

end;

A6: for x being Real st x in Z holds
ln * sec is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies ln * sec is_differentiable_in x )
assume A7: x in Z ; :: thesis: ln * sec is_differentiable_in x
then A8: sec is_differentiable_in x by A5;
( cos . x <> 0 & sec . x > 0 ) by A2, A3, A7;
hence ln * sec is_differentiable_in x by A8, TAYLOR_1:20; :: thesis: verum

end;

then A9: ln * sec is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((ln * sec ) `| Z) . x = (sin . x) / (cos . x)

proof

let x be Real; :: thesis: ( x in Z implies ((ln * sec ) `| Z) . x = (sin . x) / (cos . x) )
assume A10: x in Z ; :: thesis: ((ln * sec ) `| Z) . x = (sin . x) / (cos . x)
then A11: sec is_differentiable_in x by A5;
A12: ( cos . x <> 0 & sec . x > 0 ) by A2, A3, A10;
then diff (ln * sec ),x = (diff sec ,x) / (sec . x) by A11, TAYLOR_1:20
.= ((sin . x) / ((cos . x) ^2 )) / (sec . x) by A12, Th1
.= ((sin . x) / ((cos . x) ^2 )) / ((cos . x) " ) by A4, A10, RFUNCT_1:def 8
.= ((sin . x) * (cos . x)) / ((cos . x) * (cos . x))
.= (sin . x) / (cos . x) by A12, XCMPLX_1:92 ;
hence ((ln * sec ) `| Z) . x = (sin . x) / (cos . x) by A9, A10, FDIFF_1:def 8; :: thesis: verum

end;

hence ( ln * sec is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * sec ) `| Z) . x = (sin . x) / (cos . x) ) ) by A1, A6, FDIFF_1:16; :: thesis: verum