let a be Real; :: thesis: for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL ,REAL st Z c= dom (ln * (f1 + f2)) & f2 = #Z 3 & ( for x being Real st x in Z holds
( f1 . x = a & (f1 + f2) . x > 0 ) ) holds
( ln * (f1 + f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 + f2)) `| Z) . x = (3 * (x |^ 2)) / (a + (x |^ 3)) ) )
let Z be open Subset of REAL ; :: thesis: for f1, f2 being PartFunc of REAL ,REAL st Z c= dom (ln * (f1 + f2)) & f2 = #Z 3 & ( for x being Real st x in Z holds
( f1 . x = a & (f1 + f2) . x > 0 ) ) holds
( ln * (f1 + f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 + f2)) `| Z) . x = (3 * (x |^ 2)) / (a + (x |^ 3)) ) )
let f1, f2 be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (ln * (f1 + f2)) & f2 = #Z 3 & ( for x being Real st x in Z holds
( f1 . x = a & (f1 + f2) . x > 0 ) ) implies ( ln * (f1 + f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 + f2)) `| Z) . x = (3 * (x |^ 2)) / (a + (x |^ 3)) ) ) )
assume that
A1:
Z c= dom (ln * (f1 + f2))
and
A2:
f2 = #Z 3
and
A3:
for x being Real st x in Z holds
( f1 . x = a & (f1 + f2) . x > 0 )
; :: thesis: ( ln * (f1 + f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 + f2)) `| Z) . x = (3 * (x |^ 2)) / (a + (x |^ 3)) ) )
for y being set st y in Z holds
y in dom (f1 + f2)
by A1, FUNCT_1:21;
then A4:
Z c= dom (f1 + f2)
by TARSKI:def 3;
A5:
for x being Real st x in Z holds
f1 . x = a
by A3;
then A6:
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 3 * (x |^ 2) ) )
by A2, A4, Th20;
A7:
for x being Real st x in Z holds
ln * (f1 + f2) is_differentiable_in x
then A10:
ln * (f1 + f2) is_differentiable_on Z
by A1, FDIFF_1:16;
for x being Real st x in Z holds
((ln * (f1 + f2)) `| Z) . x = (3 * (x |^ 2)) / (a + (x |^ 3))
proof
let x be
Real;
:: thesis: ( x in Z implies ((ln * (f1 + f2)) `| Z) . x = (3 * (x |^ 2)) / (a + (x |^ 3)) )
assume A11:
x in Z
;
:: thesis: ((ln * (f1 + f2)) `| Z) . x = (3 * (x |^ 2)) / (a + (x |^ 3))
then A12:
(
f1 + f2 is_differentiable_in x &
((f1 + f2) `| Z) . x = 3
* (x |^ 2) )
by A6, FDIFF_1:16;
A13:
(f1 + f2) . x > 0
by A3, A11;
A14:
x in dom (f1 + f2)
by A1, A11, FUNCT_1:21;
diff (ln * (f1 + f2)),
x =
(diff (f1 + f2),x) / ((f1 + f2) . x)
by A12, A13, TAYLOR_1:20
.=
(((f1 + f2) `| Z) . x) / ((f1 + f2) . x)
by A6, A11, FDIFF_1:def 8
.=
(3 * (x |^ 2)) / ((f1 + f2) . x)
by A2, A4, A5, A11, Th20
.=
(3 * (x |^ 2)) / ((f1 . x) + (f2 . x))
by A14, VALUED_1:def 1
.=
(3 * (x |^ 2)) / (a + (f2 . x))
by A3, A11
.=
(3 * (x |^ 2)) / (a + (x #Z 3))
by A2, TAYLOR_1:def 1
.=
(3 * (x |^ 2)) / (a + (x |^ 3))
by PREPOWER:46
;
hence
((ln * (f1 + f2)) `| Z) . x = (3 * (x |^ 2)) / (a + (x |^ 3))
by A10, A11, FDIFF_1:def 8;
:: thesis: verum
end;
hence
( ln * (f1 + f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 + f2)) `| Z) . x = (3 * (x |^ 2)) / (a + (x |^ 3)) ) )
by A1, A7, FDIFF_1:16; :: thesis: verum