let n be natural number ; :: thesis: for Z being open Subset of REAL st Z c= dom ((#Z n) * cosec ) & 1 <= n holds
( (#Z n) * cosec is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z n) * cosec ) `| Z) . x = - ((n * (cos . x)) / ((sin . x) #Z (n + 1))) ) )
let Z be open Subset of REAL ; :: thesis: ( Z c= dom ((#Z n) * cosec ) & 1 <= n implies ( (#Z n) * cosec is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z n) * cosec ) `| Z) . x = - ((n * (cos . x)) / ((sin . x) #Z (n + 1))) ) ) )
assume A1: ( Z c= dom ((#Z n) * cosec ) & 1 <= n ) ; :: thesis: ( (#Z n) * cosec is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z n) * cosec ) `| Z) . x = - ((n * (cos . x)) / ((sin . x) #Z (n + 1))) ) )
A2: for x being Real st x in Z holds
sin . x <> 0

proof

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_1:21;
hence sin . x <> 0 by RFUNCT_1:13; :: thesis: verum

end;

dom ((#Z n) * cosec ) c= dom cosec by RELAT_1:44;
then A3: Z c= dom cosec by A1, XBOOLE_1:1;
A4: for x being Real st x in Z holds
(#Z n) * cosec is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies (#Z n) * cosec is_differentiable_in x )
assume x in Z ; :: thesis: (#Z n) * cosec is_differentiable_in x
then sin . x <> 0 by A2;
then cosec is_differentiable_in x by Th2;
hence (#Z n) * cosec is_differentiable_in x by TAYLOR_1:3; :: thesis: verum

end;

then A5: (#Z n) * cosec is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
(((#Z n) * cosec ) `| Z) . x = - ((n * (cos . x)) / ((sin . x) #Z (n + 1)))

proof

let x be Real; :: thesis: ( x in Z implies (((#Z n) * cosec ) `| Z) . x = - ((n * (cos . x)) / ((sin . x) #Z (n + 1))) )
assume A6: x in Z ; :: thesis: (((#Z n) * cosec ) `| Z) . x = - ((n * (cos . x)) / ((sin . x) #Z (n + 1)))
then A7: sin . x <> 0 by A2;
then A8: cosec is_differentiable_in x by Th2;
consider m being Nat such that
A9: n = m + 1 by A1, NAT_1:6;
set m = n - 1;
(((#Z n) * cosec ) `| Z) . x = diff ((#Z n) * cosec ),x by A5, A6, FDIFF_1:def 8
.= (n * ((cosec . x) #Z (n - 1))) * (diff cosec ,x) by A8, TAYLOR_1:3
.= (n * ((cosec . x) #Z (n - 1))) * (- ((cos . x) / ((sin . x) ^2 ))) by A7, Th2
.= (n * ((1 / (sin . x)) #Z (n - 1))) * (- ((cos . x) / ((sin . x) ^2 ))) by A3, A6, RFUNCT_1:def 8
.= (n * (1 / ((sin . x) #Z (n - 1)))) * (- ((cos . x) / ((sin . x) ^2 ))) by A9, Th3
.= (n / ((sin . x) #Z (n - 1))) * ((- (cos . x)) / ((sin . x) ^2 ))
.= (n * (- (cos . x))) / (((sin . x) #Z (n - 1)) * ((sin . x) ^2 )) by XCMPLX_1:77
.= (n * (- (cos . x))) / (((sin . x) #Z (n - 1)) * ((sin . x) #Z 2)) by FDIFF_7:1
.= (n * (- (cos . x))) / ((sin . x) #Z ((n - 1) + 2)) by A2, A6, PREPOWER:54
.= (n * (- (cos . x))) / ((sin . x) #Z (n + 1)) ;
hence (((#Z n) * cosec ) `| Z) . x = - ((n * (cos . x)) / ((sin . x) #Z (n + 1))) ; :: thesis: verum

end;

hence ( (#Z n) * cosec is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z n) * cosec ) `| Z) . x = - ((n * (cos . x)) / ((sin . x) #Z (n + 1))) ) ) by A1, A4, FDIFF_1:16; :: thesis: verum