let Z be open Subset of REAL ; :: thesis: ( Z c= dom (tan * ln ) implies ( tan * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * ln ) `| Z) . x = 1 / (x * ((cos . (ln . x)) ^2 )) ) ) )
assume A1: Z c= dom (tan * ln ) ; :: thesis: ( tan * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * ln ) `| Z) . x = 1 / (x * ((cos . (ln . x)) ^2 )) ) )
dom (tan * 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
cos . (ln . x) <> 0

proof

let x be Real; :: thesis: ( x in Z implies cos . (ln . x) <> 0 )
assume x in Z ; :: thesis: cos . (ln . x) <> 0
then ln . x in dom (sin / cos ) by A1, FUNCT_1:21;
hence cos . (ln . x) <> 0 by Th1; :: 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
tan * ln is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies tan * ln is_differentiable_in x )
assume A7: x in Z ; :: thesis: tan * ln is_differentiable_in x
then A8: ln is_differentiable_in x by A5;
( x > 0 & cos . (ln . x) <> 0 ) by A3, A4, A7;
then tan is_differentiable_in ln . x by FDIFF_7:46;
hence tan * ln is_differentiable_in x by A8, FDIFF_2:13; :: thesis: verum

end;

then A9: tan * 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
((tan * ln ) `| Z) . x = 1 / (x * ((cos . (ln . x)) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((tan * ln ) `| Z) . x = 1 / (x * ((cos . (ln . x)) ^2 )) )
assume A11: x in Z ; :: thesis: ((tan * ln ) `| Z) . x = 1 / (x * ((cos . (ln . x)) ^2 ))
then A12: ln is_differentiable_in x by A5;
A13: ( x > 0 & cos . (ln . x) <> 0 ) by A3, A4, A11;
then tan is_differentiable_in ln . x by FDIFF_7:46;
then diff (tan * ln ),x = (diff tan ,(ln . x)) * (diff ln ,x) by A12, FDIFF_2:13
.= (1 / ((cos . (ln . x)) ^2 )) * (diff ln ,x) by A13, FDIFF_7:46
.= (1 / x) / ((cos . (ln . x)) ^2 ) by A10, A11
.= 1 / (x * ((cos . (ln . x)) ^2 )) by XCMPLX_1:79 ;
hence ((tan * ln ) `| Z) . x = 1 / (x * ((cos . (ln . x)) ^2 )) by A9, A11, FDIFF_1:def 8; :: thesis: verum

end;

hence ( tan * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * ln ) `| Z) . x = 1 / (x * ((cos . (ln . x)) ^2 )) ) ) by A1, A6, FDIFF_1:16; :: thesis: verum