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: Z c= dom (ln * (f1 + ((- 1) (#) f2))) by A1, VALUED_1:8;
A5: for x being Real st x in Z holds
( f1 . x = (a ^2 ) + (0 * x) & (f1 + ((- 1) (#) f2)) . x > 0 ) by A3;
then A6: ( 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, A4, 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 A7: x in Z ; :: thesis: ((- (ln * (f1 - f2))) `| Z) . x = (2 * x) / ((a ^2 ) - (x |^ 2))
then ((- (ln * (f1 - f2))) `| Z) . x = (- 1) * (diff (ln * (f1 + ((- 1) (#) f2))),x) by A1, A6, FDIFF_1:28
.= (- 1) * (((ln * (f1 + ((- 1) (#) f2))) `| Z) . x) by A6, A7, FDIFF_1:def 8
.= (- 1) * ((0 + ((2 * (- 1)) * x)) / (((a ^2 ) + (0 * x)) + ((- 1) * (x |^ 2)))) by A2, A4, A5, A7, Th13
.= ((- 1) * ((2 * (- 1)) * x)) / ((a ^2 ) + ((- 1) * (x |^ 2))) by XCMPLX_1:75
.= (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 A1, A6, FDIFF_1:28; :: thesis: verum