let c, a, b be Real; :: thesis: for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL ,REAL st Z c= dom (ln * (f1 + (c (#) f2))) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a + (b * x) & (f1 + (c (#) f2)) . x > 0 ) ) holds
( ln * (f1 + (c (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 + (c (#) f2))) `| Z) . x = (b + ((2 * c) * x)) / ((a + (b * x)) + (c * (x |^ 2))) ) )
let Z be open Subset of REAL ; :: thesis: for f1, f2 being PartFunc of REAL ,REAL st Z c= dom (ln * (f1 + (c (#) f2))) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a + (b * x) & (f1 + (c (#) f2)) . x > 0 ) ) holds
( ln * (f1 + (c (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 + (c (#) f2))) `| Z) . x = (b + ((2 * c) * x)) / ((a + (b * x)) + (c * (x |^ 2))) ) )
let f1, f2 be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (ln * (f1 + (c (#) f2))) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a + (b * x) & (f1 + (c (#) f2)) . x > 0 ) ) implies ( ln * (f1 + (c (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 + (c (#) f2))) `| Z) . x = (b + ((2 * c) * x)) / ((a + (b * x)) + (c * (x |^ 2))) ) ) )
assume A1:
( Z c= dom (ln * (f1 + (c (#) f2))) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a + (b * x) & (f1 + (c (#) f2)) . x > 0 ) ) )
; :: thesis: ( ln * (f1 + (c (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 + (c (#) f2))) `| Z) . x = (b + ((2 * c) * x)) / ((a + (b * x)) + (c * (x |^ 2))) ) )
then
for y being set st y in Z holds
y in dom (f1 + (c (#) f2))
by FUNCT_1:21;
then A2:
Z c= dom (f1 + (c (#) f2))
by TARSKI:def 3;
then
Z c= (dom f1) /\ (dom (c (#) f2))
by VALUED_1:def 1;
then A3:
( Z c= dom f1 & Z c= dom (c (#) f2) )
by XBOOLE_1:18;
A4:
( ( for x being Real st x in Z holds
f1 . x = a + (b * x) ) & f2 = #Z 2 )
by A1;
then A5:
( f1 + (c (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + (c (#) f2)) `| Z) . x = b + ((2 * c) * x) ) )
by A2, Th12;
A6:
for x being Real st x in Z holds
ln * (f1 + (c (#) f2)) is_differentiable_in x
then A9:
ln * (f1 + (c (#) f2)) is_differentiable_on Z
by A1, FDIFF_1:16;
for x being Real st x in Z holds
((ln * (f1 + (c (#) f2))) `| Z) . x = (b + ((2 * c) * x)) / ((a + (b * x)) + (c * (x |^ 2)))
proof
let x be
Real;
:: thesis: ( x in Z implies ((ln * (f1 + (c (#) f2))) `| Z) . x = (b + ((2 * c) * x)) / ((a + (b * x)) + (c * (x |^ 2))) )
assume A10:
x in Z
;
:: thesis: ((ln * (f1 + (c (#) f2))) `| Z) . x = (b + ((2 * c) * x)) / ((a + (b * x)) + (c * (x |^ 2)))
then A11:
x in dom (f1 + (c (#) f2))
by A1, FUNCT_1:21;
A12:
f1 + (c (#) f2) is_differentiable_in x
by A5, A10, FDIFF_1:16;
A13:
(
f1 . x = a + (b * x) &
(f1 + (c (#) f2)) . x > 0 )
by A1, A10;
A14:
(f1 + (c (#) f2)) . x =
(f1 . x) + ((c (#) f2) . x)
by A11, VALUED_1:def 1
.=
(f1 . x) + (c * (f2 . x))
by A3, A10, VALUED_1:def 5
.=
(a + (b * x)) + (c * (f2 . x))
by A1, A10
.=
(a + (b * x)) + (c * (x #Z 2))
by A1, TAYLOR_1:def 1
.=
(a + (b * x)) + (c * (x |^ 2))
by PREPOWER:46
;
diff (ln * (f1 + (c (#) f2))),
x =
(diff (f1 + (c (#) f2)),x) / ((f1 + (c (#) f2)) . x)
by A12, A13, TAYLOR_1:20
.=
(((f1 + (c (#) f2)) `| Z) . x) / ((f1 + (c (#) f2)) . x)
by A5, A10, FDIFF_1:def 8
.=
(b + ((2 * c) * x)) / ((a + (b * x)) + (c * (x |^ 2)))
by A2, A4, A10, A14, Th12
;
hence
((ln * (f1 + (c (#) f2))) `| Z) . x = (b + ((2 * c) * x)) / ((a + (b * x)) + (c * (x |^ 2)))
by A9, A10, FDIFF_1:def 8;
:: thesis: verum
end;
hence
( ln * (f1 + (c (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 + (c (#) f2))) `| Z) . x = (b + ((2 * c) * x)) / ((a + (b * x)) + (c * (x |^ 2))) ) )
by A1, A6, FDIFF_1:16; :: thesis: verum