let Z be open Subset of REAL; ( Z c= dom (arctan * tan) & ( for x being Real st x in Z holds
( tan . x > - 1 & tan . x < 1 ) ) implies ( arctan * tan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * tan) `| Z) . x = 1 ) ) )
assume that
A1:
Z c= dom (arctan * tan)
and
A2:
for x being Real st x in Z holds
( tan . x > - 1 & tan . x < 1 )
; ( arctan * tan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * tan) `| Z) . x = 1 ) )
dom (arctan * tan) c= dom tan
by RELAT_1:25;
then A3:
Z c= dom tan
by A1, XBOOLE_1:1;
A4:
for x being Real st x in Z holds
arctan * tan is_differentiable_in x
then A7:
arctan * tan is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((arctan * tan) `| Z) . x = 1
proof
let x be
Real;
( x in Z implies ((arctan * tan) `| Z) . x = 1 )
assume A8:
x in Z
;
((arctan * tan) `| Z) . x = 1
then A9:
(
tan . x > - 1 &
tan . x < 1 )
by A2;
A10:
tan . x = (sin . x) / (cos . x)
by A3, A8, RFUNCT_1:def 1;
A11:
cos . x <> 0
by A3, A8, FDIFF_8:1;
then A12:
tan is_differentiable_in x
by FDIFF_7:46;
A13:
(cos . x) ^2 <> 0
by A11, SQUARE_1:12;
((arctan * tan) `| Z) . x =
diff (
(arctan * tan),
x)
by A7, A8, FDIFF_1:def 7
.=
(diff (tan,x)) / (1 + ((tan . x) ^2))
by A12, A9, SIN_COS9:85
.=
(1 / ((cos . x) ^2)) / (1 + ((tan . x) ^2))
by A11, FDIFF_7:46
.=
1
/ (((cos . x) ^2) * (1 + (((sin . x) / (cos . x)) * ((sin . x) / (cos . x)))))
by A10, XCMPLX_1:78
.=
1
/ (((cos . x) ^2) * (1 + (((sin . x) ^2) / ((cos . x) ^2))))
by XCMPLX_1:76
.=
1
/ (((cos . x) ^2) + ((((cos . x) ^2) * ((sin . x) ^2)) / ((cos . x) ^2)))
.=
1
/ (((cos . x) ^2) + ((sin . x) ^2))
by A13, XCMPLX_1:89
.=
1
/ 1
by SIN_COS:28
.=
1
;
hence
((arctan * tan) `| Z) . x = 1
;
verum
end;
hence
( arctan * tan is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * tan) `| Z) . x = 1 ) )
by A1, A4, FDIFF_1:9; verum