let a be Real; :: thesis: for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (4 * (a ^2))) (#) f) & f = ln * ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & (f1 - f2) . x > 0 & a <> 0 ) ) holds
( (1 / (4 * (a ^2))) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) ) )
let Z be open Subset of REAL; :: thesis: for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (4 * (a ^2))) (#) f) & f = ln * ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & (f1 - f2) . x > 0 & a <> 0 ) ) holds
( (1 / (4 * (a ^2))) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) ) )
let f, f1, f2 be PartFunc of REAL,REAL; :: thesis: ( Z c= dom ((1 / (4 * (a ^2))) (#) f) & f = ln * ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & (f1 - f2) . x > 0 & a <> 0 ) ) implies ( (1 / (4 * (a ^2))) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) ) ) )
assume that
A1: Z c= dom ((1 / (4 * (a ^2))) (#) f) and
A2: ( f = ln * ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 ) and
A3: for x being Real st x in Z holds
( f1 . x = a ^2 & (f1 - f2) . x > 0 & a <> 0 ) ; :: thesis: ( (1 / (4 * (a ^2))) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) ) )
A4: Z c= dom f by A1, VALUED_1:def 5;
then A5: f is_differentiable_on Z by A2, A3, Th5;
for x being Real st x in Z holds
(((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4))

proof

let x be Real; :: thesis: ( x in Z implies (((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) )
assume A6: x in Z ; :: thesis: (((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4))
then a <> 0 by A3;
then a ^2 > 0 by SQUARE_1:12;
then A7: 4 * (a ^2) > 4 * 0 by XREAL_1:68;
(((1 / (4 * (a ^2))) (#) f) `| Z) . x = (1 / (4 * (a ^2))) * (diff (f,x)) by A1, A5, A6, FDIFF_1:20
.= (1 / (4 * (a ^2))) * ((f `| Z) . x) by A5, A6, FDIFF_1:def 7
.= (1 / (4 * (a ^2))) * (((4 * (a ^2)) * x) / ((a |^ 4) - (x |^ 4))) by A2, A3, A4, A6, Th5
.= (1 / (4 * (a ^2))) * ((4 * (a ^2)) * (x / ((a |^ 4) - (x |^ 4)))) by XCMPLX_1:74
.= (x / ((a |^ 4) - (x |^ 4))) * ((1 / (4 * (a ^2))) * (4 * (a ^2)))
.= x / ((a |^ 4) - (x |^ 4)) by A7, XCMPLX_1:108 ;
hence (((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) ; :: thesis: verum

end;

hence ( (1 / (4 * (a ^2))) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) ) ) by A1, A5, FDIFF_1:20; :: thesis: verum