let Z be open Subset of REAL ; :: thesis: ( Z c= dom (arctan * arctan ) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
( arctan . x > - 1 & arctan . x < 1 ) ) implies ( arctan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * arctan ) `| Z) . x = 1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 ))) ) ) )
assume that
A1:
Z c= dom (arctan * arctan )
and
A2:
Z c= ].(- 1),1.[
and
A3:
for x being Real st x in Z holds
( arctan . x > - 1 & arctan . x < 1 )
; :: thesis: ( arctan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * arctan ) `| Z) . x = 1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 ))) ) )
AA:
for x being Real st x in Z holds
arctan * arctan is_differentiable_in x
then A6:
arctan * arctan is_differentiable_on Z
by A1, FDIFF_1:16;
for x being Real st x in Z holds
((arctan * arctan ) `| Z) . x = 1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 )))
proof
let x be
Real;
:: thesis: ( x in Z implies ((arctan * arctan ) `| Z) . x = 1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 ))) )
assume A7:
x in Z
;
:: thesis: ((arctan * arctan ) `| Z) . x = 1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 )))
A8:
arctan is_differentiable_on Z
by A2, SIN_COS9:81;
then A9:
arctan is_differentiable_in x
by A7, FDIFF_1:16;
A10:
(
arctan . x > - 1 &
arctan . x < 1 )
by A3, A7;
((arctan * arctan ) `| Z) . x =
diff (arctan * arctan ),
x
by A6, A7, FDIFF_1:def 8
.=
(diff arctan ,x) / (1 + ((arctan . x) ^2 ))
by A9, A10, SIN_COS9:85
.=
((arctan `| Z) . x) / (1 + ((arctan . x) ^2 ))
by A7, A8, FDIFF_1:def 8
.=
(1 / (1 + (x ^2 ))) / (1 + ((arctan . x) ^2 ))
by A2, A7, SIN_COS9:81
.=
1
/ ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 )))
by XCMPLX_1:79
;
hence
((arctan * arctan ) `| Z) . x = 1
/ ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 )))
;
:: thesis: verum
end;
hence
( arctan * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * arctan ) `| Z) . x = 1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 ))) ) )
by A1, AA, FDIFF_1:16; :: thesis: verum