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)) ) )
A4: f2 = 1 (#) f2 by RFUNCT_1:33;
A5: Z c= dom (ln * (f1 + (1 (#) f2))) by A1, RFUNCT_1:33;
A6: for x being Real st x in Z holds
( f1 . x = (a ^2 ) + (0 * x) & (f1 + (1 (#) f2)) . x > 0 ) by A3, A4;
then A7: ( ln * (f1 + (1 (#) f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 + (1 (#) f2))) `| Z) . x = (0 + ((2 * 1) * x)) / (((a ^2 ) + (0 * x)) + (1 * (x |^ 2))) ) ) by A2, A5, Th13;
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 A8: 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_1:33
.= (0 + ((2 * 1) * x)) / (((a ^2 ) + (0 * x)) + (1 * (x |^ 2))) by A2, A5, A6, A8, Th13
.= (2 * x) / ((a ^2 ) + (x |^ 2)) ;
hence ((ln * (f1 + f2)) `| Z) . x = (2 * x) / ((a ^2 ) + (x |^ 2)) ; :: 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 = (2 * x) / ((a ^2 ) + (x |^ 2)) ) ) by A7, RFUNCT_1:33; :: thesis: verum