let Z be open Subset of REAL ; :: thesis: ( Z c= dom (cosec * ln ) implies ( cosec * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * ln ) `| Z) . x = - ((cos . (ln . x)) / (x * ((sin . (ln . x)) ^2 ))) ) ) )
assume A1: Z c= dom (cosec * ln ) ; :: thesis: ( cosec * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * ln ) `| Z) . x = - ((cos . (ln . x)) / (x * ((sin . (ln . x)) ^2 ))) ) )
dom (cosec * ln ) c= dom ln by RELAT_1:44;
then A2: Z c= dom ln by A1, XBOOLE_1:1;
A3: for x being Real st x in Z holds
x > 0

proof

let x be Real; :: thesis: ( x in Z implies x > 0 )
assume x in Z ; :: thesis: x > 0
then x in right_open_halfline 0 by A2, TAYLOR_1:18;
then ex g being Real st
( x = g & 0 < g ) by Lm1;
hence x > 0 ; :: thesis: verum

end;

A4: for x being Real st x in Z holds
sin . (ln . x) <> 0

proof

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

end;

A5: for x being Real st x in Z holds
ln is_differentiable_in x by A3, TAYLOR_1:18;
A6: for x being Real st x in Z holds
cosec * ln is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cosec * ln is_differentiable_in x )
assume A7: x in Z ; :: thesis: cosec * ln is_differentiable_in x
then A8: ln is_differentiable_in x by A5;
( x > 0 & sin . (ln . x) <> 0 ) by A3, A4, A7;
then cosec is_differentiable_in ln . x by Th2;
hence cosec * ln is_differentiable_in x by A8, FDIFF_2:13; :: thesis: verum

end;

then A9: cosec * ln is_differentiable_on Z by A1, FDIFF_1:16;
A10: for x being Real st x in Z holds
diff ln ,x = 1 / x

proof

let x be Real; :: thesis: ( x in Z implies diff ln ,x = 1 / x )
assume x in Z ; :: thesis: diff ln ,x = 1 / x
then x > 0 by A3;
then x in right_open_halfline 0 by Lm1;
hence diff ln ,x = 1 / x by TAYLOR_1:18; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies ((cosec * ln ) `| Z) . x = - ((cos . (ln . x)) / (x * ((sin . (ln . x)) ^2 ))) )
assume A11: x in Z ; :: thesis: ((cosec * ln ) `| Z) . x = - ((cos . (ln . x)) / (x * ((sin . (ln . x)) ^2 )))
then A12: ln is_differentiable_in x by A5;
A13: ( x > 0 & sin . (ln . x) <> 0 ) by A3, A4, A11;
then cosec is_differentiable_in ln . x by Th2;
then diff (cosec * ln ),x = (diff cosec ,(ln . x)) * (diff ln ,x) by A12, FDIFF_2:13
.= (- ((cos . (ln . x)) / ((sin . (ln . x)) ^2 ))) * (diff ln ,x) by A13, Th2
.= (1 / x) * ((- (cos . (ln . x))) / ((sin . (ln . x)) ^2 )) by A10, A11
.= (1 * (- (cos . (ln . x)))) / (x * ((sin . (ln . x)) ^2 )) by XCMPLX_1:77 ;
hence ((cosec * ln ) `| Z) . x = - ((cos . (ln . x)) / (x * ((sin . (ln . x)) ^2 ))) by A9, A11, FDIFF_1:def 8; :: thesis: verum

end;

hence ( cosec * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * ln ) `| Z) . x = - ((cos . (ln . x)) / (x * ((sin . (ln . x)) ^2 ))) ) ) by A1, A6, FDIFF_1:16; :: thesis: verum