let a, b, c 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 that
A1: Z c= dom (ln * (f1 + (c (#) f2))) and
A2: f2 = #Z 2 and
A3: 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))) ) )
for y being object st y in Z holds
y in dom (f1 + (c (#) f2)) by A1, FUNCT_1:11;
then A4: Z c= dom (f1 + (c (#) f2)) by TARSKI:def 3;
then Z c= (dom f1) /\ (dom (c (#) f2)) by VALUED_1:def 1;
then A5: Z c= dom (c (#) f2) by XBOOLE_1:18;
A6: for x being Real st x in Z holds
f1 . x = a + (b * x) by A3;
then A7: f1 + (c (#) f2) is_differentiable_on Z by A2, A4, Th12;
A8: for x being Real st x in Z holds
ln * (f1 + (c (#) f2)) is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies ln * (f1 + (c (#) f2)) is_differentiable_in x )
assume x in Z ; :: thesis: ln * (f1 + (c (#) f2)) is_differentiable_in x
then ( f1 + (c (#) f2) is_differentiable_in x & (f1 + (c (#) f2)) . x > 0 ) by A3, A7, FDIFF_1:9;
hence ln * (f1 + (c (#) f2)) is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

end;

then A9: ln * (f1 + (c (#) f2)) is_differentiable_on Z by A1, FDIFF_1:9;
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 x in dom (f1 + (c (#) f2)) by A1, FUNCT_1:11;
then A11: (f1 + (c (#) f2)) . x = (f1 . x) + ((c (#) f2) . x) by VALUED_1:def 1
.= (f1 . x) + (c * (f2 . x)) by A5, A10, VALUED_1:def 5
.= (a + (b * x)) + (c * (f2 . x)) by A3, A10
.= (a + (b * x)) + (c * (x #Z 2)) by A2, TAYLOR_1:def 1
.= (a + (b * x)) + (c * (x |^ 2)) by PREPOWER:36 ;
( f1 + (c (#) f2) is_differentiable_in x & (f1 + (c (#) f2)) . x > 0 ) by A3, A7, A10, FDIFF_1:9;
then diff ((ln * (f1 + (c (#) f2))),x) = (diff ((f1 + (c (#) f2)),x)) / ((f1 + (c (#) f2)) . x) by TAYLOR_1:20
.= (((f1 + (c (#) f2)) `| Z) . x) / ((f1 + (c (#) f2)) . x) by A7, A10, FDIFF_1:def 7
.= (b + ((2 * c) * x)) / ((a + (b * x)) + (c * (x |^ 2))) by A2, A4, A6, A10, A11, Th12 ;
hence ((ln * (f1 + (c (#) f2))) `| Z) . x = (b + ((2 * c) * x)) / ((a + (b * x)) + (c * (x |^ 2))) by A9, A10, FDIFF_1:def 7; :: 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, A8, FDIFF_1:9; :: thesis: verum