let r, s, h be Real; for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (arctan * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds
( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = r + (s * x) ) & f2 = #Z 2 holds
( arctan * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) )
let Z be open Subset of REAL; for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (arctan * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds
( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = r + (s * x) ) & f2 = #Z 2 holds
( arctan * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) )
let f, f1, f2 be PartFunc of REAL,REAL; ( Z c= dom (arctan * f) & f = f1 + (h (#) f2) & ( for x being Real st x in Z holds
( f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = r + (s * x) ) & f2 = #Z 2 implies ( arctan * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) ) )
assume that
A1:
Z c= dom (arctan * f)
and
A2:
f = f1 + (h (#) f2)
and
A3:
for x being Real st x in Z holds
( f . x > - 1 & f . x < 1 )
and
A4:
for x being Real st x in Z holds
f1 . x = r + (s * x)
and
A5:
f2 = #Z 2
; ( arctan * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) )
dom (arctan * f) c= dom f
by RELAT_1:25;
then A6:
Z c= dom (f1 + (h (#) f2))
by A1, A2;
then
Z c= (dom f1) /\ (dom (h (#) f2))
by VALUED_1:def 1;
then A7:
Z c= dom (h (#) f2)
by XBOOLE_1:18;
A8:
f1 + (h (#) f2) is_differentiable_on Z
by A4, A5, A6, FDIFF_4:12;
A9:
for x being Real st x in Z holds
arctan * (f1 + (h (#) f2)) is_differentiable_in x
then A13:
arctan * (f1 + (h (#) f2)) is_differentiable_on Z
by A1, A2, FDIFF_1:9;
for x being Real st x in Z holds
((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))
proof
let x be
Real;
( x in Z implies ((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) )
assume A14:
x in Z
;
((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))
then A15:
(f1 + (h (#) f2)) . x =
(f1 . x) + ((h (#) f2) . x)
by A6, VALUED_1:def 1
.=
(f1 . x) + (h * (f2 . x))
by A7, A14, VALUED_1:def 5
.=
(r + (s * x)) + (h * (f2 . x))
by A4, A14
.=
(r + (s * x)) + (h * (x #Z (1 + 1)))
by A5, TAYLOR_1:def 1
.=
(r + (s * x)) + (h * ((x #Z 1) * (x #Z 1)))
by TAYLOR_1:1
.=
(r + (s * x)) + (h * (x * (x #Z 1)))
by PREPOWER:35
.=
(r + (s * x)) + (h * (x ^2))
by PREPOWER:35
;
A16:
f is_differentiable_in x
by A2, A8, A14, FDIFF_1:9;
A17:
f . x > - 1
by A3, A14;
A18:
f . x < 1
by A3, A14;
((arctan * (f1 + (h (#) f2))) `| Z) . x =
diff (
(arctan * f),
x)
by A2, A13, A14, FDIFF_1:def 7
.=
(diff (f,x)) / (1 + ((f . x) ^2))
by A16, A17, A18, Th85
.=
(((f1 + (h (#) f2)) `| Z) . x) / (1 + ((f . x) ^2))
by A2, A8, A14, FDIFF_1:def 7
.=
(s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))
by A2, A4, A5, A6, A14, A15, FDIFF_4:12
;
hence
((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2))
;
verum
end;
hence
( arctan * (f1 + (h (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * (f1 + (h (#) f2))) `| Z) . x = (s + ((2 * h) * x)) / (1 + (((r + (s * x)) + (h * (x ^2))) ^2)) ) )
by A1, A2, A9, FDIFF_1:9; verum