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

sec . x > 0

then A3: Z c= dom sec by A1, XBOOLE_1:1;

A4: for x being Real st x in Z holds

cos . x <> 0

sec is_differentiable_in x

ln * sec is_differentiable_in x

for x being Real st x in Z holds

((ln * sec) `| Z) . x = (sin . x) / (cos . x)

((ln * sec) `| Z) . x = (sin . x) / (cos . x) ) ) by A1, A6, FDIFF_1:9; :: thesis: verum

((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

sec . x > 0

proof

dom (ln * sec) c= dom sec
by RELAT_1:25;
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_1:11, TAYLOR_1:18;

then ex g being Real st

( sec . x = g & 0 < g ) by Lm1;

hence sec . x > 0 ; :: thesis: verum

end;assume x in Z ; :: thesis: sec . x > 0

then sec . x in right_open_halfline 0 by A1, FUNCT_1:11, TAYLOR_1:18;

then ex g being Real st

( sec . x = g & 0 < g ) by Lm1;

hence sec . x > 0 ; :: thesis: verum

then A3: Z c= dom sec by A1, XBOOLE_1:1;

A4: for x being Real st x in Z holds

cos . x <> 0

proof

A5:
for x being Real st x in Z holds
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_1:11;

hence cos . x <> 0 by RFUNCT_1:3; :: thesis: verum

end;assume x in Z ; :: thesis: cos . x <> 0

then x in dom sec by A1, FUNCT_1:11;

hence cos . x <> 0 by RFUNCT_1:3; :: thesis: verum

sec is_differentiable_in x

proof

A6:
for x being Real st x in Z holds
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 A4;

hence sec is_differentiable_in x by Th1; :: thesis: verum

end;assume x in Z ; :: thesis: sec is_differentiable_in x

then cos . x <> 0 by A4;

hence sec is_differentiable_in x by Th1; :: thesis: verum

ln * sec is_differentiable_in x

proof

then A7:
ln * sec is_differentiable_on Z
by A1, FDIFF_1:9;
let x be Real; :: thesis: ( x in Z implies ln * sec is_differentiable_in x )

assume x in Z ; :: thesis: ln * sec is_differentiable_in x

then ( sec is_differentiable_in x & sec . x > 0 ) by A2, A5;

hence ln * sec is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

end;assume x in Z ; :: thesis: ln * sec is_differentiable_in x

then ( sec is_differentiable_in x & sec . x > 0 ) by A2, A5;

hence ln * sec is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

for x being Real st x in Z holds

((ln * sec) `| Z) . x = (sin . x) / (cos . x)

proof

hence
( ln * sec is_differentiable_on Z & ( for x being Real st x in Z holds
let x be Real; :: thesis: ( x in Z implies ((ln * sec) `| Z) . x = (sin . x) / (cos . x) )

assume A8: x in Z ; :: thesis: ((ln * sec) `| Z) . x = (sin . x) / (cos . x)

then A9: cos . x <> 0 by A4;

( sec is_differentiable_in x & sec . x > 0 ) by A2, A5, A8;

then diff ((ln * sec),x) = (diff (sec,x)) / (sec . x) by TAYLOR_1:20

.= ((sin . x) / ((cos . x) ^2)) / (sec . x) by A9, Th1

.= ((sin . x) / ((cos . x) ^2)) / ((cos . x) ") by A3, A8, RFUNCT_1:def 2

.= ((sin . x) * (cos . x)) / ((cos . x) * (cos . x))

.= (sin . x) / (cos . x) by A4, A8, XCMPLX_1:91 ;

hence ((ln * sec) `| Z) . x = (sin . x) / (cos . x) by A7, A8, FDIFF_1:def 7; :: thesis: verum

end;assume A8: x in Z ; :: thesis: ((ln * sec) `| Z) . x = (sin . x) / (cos . x)

then A9: cos . x <> 0 by A4;

( sec is_differentiable_in x & sec . x > 0 ) by A2, A5, A8;

then diff ((ln * sec),x) = (diff (sec,x)) / (sec . x) by TAYLOR_1:20

.= ((sin . x) / ((cos . x) ^2)) / (sec . x) by A9, Th1

.= ((sin . x) / ((cos . x) ^2)) / ((cos . x) ") by A3, A8, RFUNCT_1:def 2

.= ((sin . x) * (cos . x)) / ((cos . x) * (cos . x))

.= (sin . x) / (cos . x) by A4, A8, XCMPLX_1:91 ;

hence ((ln * sec) `| Z) . x = (sin . x) / (cos . x) by A7, A8, FDIFF_1:def 7; :: thesis: verum

((ln * sec) `| Z) . x = (sin . x) / (cos . x) ) ) by A1, A6, FDIFF_1:9; :: thesis: verum