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 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & (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 = (2 * x) / ((a ^2) + (x |^ 2)) ) )
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 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & (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 = (2 * x) / ((a ^2) + (x |^ 2)) ) )
let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (ln * (f1 + f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & (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 = (2 * x) / ((a ^2) + (x |^ 2)) ) ) )
assume that
A1: Z c= dom (ln * (f1 + f2)) and
A2: f2 = #Z 2 and
A3: for x being Real st x in Z holds
( f1 . x = a ^2 & (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 = (2 * x) / ((a ^2) + (x |^ 2)) ) )
f2 = 1 (#) f2 by RFUNCT_1:21;
then A4: for x being Real st x in Z holds
( f1 . x = (a ^2) + (0 * x) & (f1 + (1 (#) f2)) . x > 0 ) by A3;
A5: Z c= dom (ln * (f1 + (1 (#) f2))) by A1, RFUNCT_1:21;
A6: for x being Real st x in Z holds
((ln * (f1 + f2)) `| Z) . x = (2 * x) / ((a ^2) + (x |^ 2))

proof

let x be Real; :: thesis: ( x in Z implies ((ln * (f1 + f2)) `| Z) . x = (2 * x) / ((a ^2) + (x |^ 2)) )
assume A7: x in Z ; :: thesis: ((ln * (f1 + f2)) `| Z) . x = (2 * x) / ((a ^2) + (x |^ 2))
((ln * (f1 + f2)) `| Z) . x = ((ln * (f1 + (1 (#) f2))) `| Z) . x by RFUNCT_1:21
.= (0 + ((2 * 1) * x)) / (((a ^2) + (0 * x)) + (1 * (x |^ 2))) by A2, A5, A4, A7, Th13
.= (2 * x) / ((a ^2) + (x |^ 2)) ;
hence ((ln * (f1 + f2)) `| Z) . x = (2 * x) / ((a ^2) + (x |^ 2)) ; :: thesis: verum

end;

ln * (f1 + (1 (#) f2)) is_differentiable_on Z by A2, A5, A4, Th13;
hence ( ln * (f1 + f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 + f2)) `| Z) . x = (2 * x) / ((a ^2) + (x |^ 2)) ) ) by A6, RFUNCT_1:21; :: thesis: verum