let Z be open Subset of REAL; :: thesis: ( Z c= dom (ln * cosec) implies ( ln * cosec is_differentiable_on Z & ( for x being Real st x in Z holds

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

assume A1: Z c= dom (ln * cosec) ; :: thesis: ( ln * cosec is_differentiable_on Z & ( for x being Real st x in Z holds

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

A2: for x being Real st x in Z holds

cosec . x > 0

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

A4: for x being Real st x in Z holds

sin . x <> 0

cosec is_differentiable_in x

ln * cosec is_differentiable_in x

for x being Real st x in Z holds

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

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

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

assume A1: Z c= dom (ln * cosec) ; :: thesis: ( ln * cosec is_differentiable_on Z & ( for x being Real st x in Z holds

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

A2: for x being Real st x in Z holds

cosec . x > 0

proof

dom (ln * cosec) c= dom cosec
by RELAT_1:25;
let x be Real; :: thesis: ( x in Z implies cosec . x > 0 )

assume x in Z ; :: thesis: cosec . x > 0

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

then ex g being Real st

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

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

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

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

then ex g being Real st

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

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

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

A4: for x being Real st x in Z holds

sin . x <> 0

proof

A5:
for x being Real st x in Z holds
let x be Real; :: thesis: ( x in Z implies sin . x <> 0 )

assume x in Z ; :: thesis: sin . x <> 0

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

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

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

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

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

cosec is_differentiable_in x

proof

A6:
for x being Real st x in Z holds
let x be Real; :: thesis: ( x in Z implies cosec is_differentiable_in x )

assume x in Z ; :: thesis: cosec is_differentiable_in x

then sin . x <> 0 by A4;

hence cosec is_differentiable_in x by Th2; :: thesis: verum

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

then sin . x <> 0 by A4;

hence cosec is_differentiable_in x by Th2; :: thesis: verum

ln * cosec is_differentiable_in x

proof

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

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

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

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

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

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

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

for x being Real st x in Z holds

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

proof

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

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

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

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

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

.= (- ((cos . x) / ((sin . x) ^2))) / (cosec . x) by A9, Th2

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

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

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

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

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

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

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

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

.= (- ((cos . x) / ((sin . x) ^2))) / (cosec . x) by A9, Th2

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

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

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

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

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