let r be Real; for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((id Z) (#) (arctan * f)) & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) holds
( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) )
let Z be open Subset of REAL; for f being PartFunc of REAL,REAL st Z c= dom ((id Z) (#) (arctan * f)) & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) holds
( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) )
let f be PartFunc of REAL,REAL; ( Z c= dom ((id Z) (#) (arctan * f)) & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) implies ( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) ) )
assume that
A1:
Z c= dom ((id Z) (#) (arctan * f))
and
A2:
for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 )
; ( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) )
A3:
Z c= (dom (id Z)) /\ (dom (arctan * f))
by A1, VALUED_1:def 4;
then A4:
Z c= dom (id Z)
by XBOOLE_1:18;
A5:
Z c= dom (arctan * f)
by A3, XBOOLE_1:18;
for x being Real st x in Z holds
f . x = ((1 / r) * x) + 0
then A6:
for x being Real st x in Z holds
( f . x = ((1 / r) * x) + 0 & f . x > - 1 & f . x < 1 )
by A2;
then A7:
arctan * f is_differentiable_on Z
by A5, Th87;
A8:
for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0
by FUNCT_1:18;
then A9:
id Z is_differentiable_on Z
by A4, FDIFF_1:23;
A10:
for x being Real st x in Z holds
((arctan * f) `| Z) . x = 1 / (r * (1 + ((x / r) ^2)))
for x being Real st x in Z holds
(((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2))))
proof
let x be
Real;
( x in Z implies (((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) )
assume A11:
x in Z
;
(((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2))))
then A12:
(arctan * f) . x =
arctan . (f . x)
by A5, FUNCT_1:12
.=
arctan . (x / r)
by A2, A11
;
(((id Z) (#) (arctan * f)) `| Z) . x =
(((arctan * f) . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff ((arctan * f),x)))
by A1, A9, A7, A11, FDIFF_1:21
.=
(((arctan * f) . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff ((arctan * f),x)))
by A9, A11, FDIFF_1:def 7
.=
(((arctan * f) . x) * 1) + (((id Z) . x) * (diff ((arctan * f),x)))
by A4, A8, A11, FDIFF_1:23
.=
(((arctan * f) . x) * 1) + (((id Z) . x) * (((arctan * f) `| Z) . x))
by A7, A11, FDIFF_1:def 7
.=
((arctan * f) . x) + (x * (((arctan * f) `| Z) . x))
by A11, FUNCT_1:18
.=
(arctan . (x / r)) + (x * (1 / (r * (1 + ((x / r) ^2)))))
by A10, A11, A12
.=
(arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2))))
;
hence
(((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2))))
;
verum
end;
hence
( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2)))) ) )
by A1, A9, A7, FDIFF_1:21; verum