let a be Real; :: thesis: for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL ,REAL st Z c= dom (f1 / f2) & ( for x being Real st x in Z holds
( f1 . x = a + x & f2 . x = a - x & f2 . x <> 0 ) ) holds
( f1 / f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2 ) ) )
let Z be open Subset of REAL ; :: thesis: for f1, f2 being PartFunc of REAL ,REAL st Z c= dom (f1 / f2) & ( for x being Real st x in Z holds
( f1 . x = a + x & f2 . x = a - x & f2 . x <> 0 ) ) holds
( f1 / f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2 ) ) )
let f1, f2 be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (f1 / f2) & ( for x being Real st x in Z holds
( f1 . x = a + x & f2 . x = a - x & f2 . x <> 0 ) ) implies ( f1 / f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2 ) ) ) )
assume A1: ( Z c= dom (f1 / f2) & ( for x being Real st x in Z holds
( f1 . x = a + x & f2 . x = a - x & f2 . x <> 0 ) ) ) ; :: thesis: ( f1 / f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2 ) ) )
then A2: Z c= (dom f1) /\ ((dom f2) \ (f2 " {0 })) by RFUNCT_1:def 4;
then A3: ( Z c= dom f1 & Z c= (dom f2) \ (f2 " {0 }) ) by XBOOLE_1:18;
A4: Z c= dom f2 by A2, XBOOLE_1:1;
A5: for x being Real st x in Z holds
f1 . x = (1 * x) + a by A1;
then A6: ( f1 is_differentiable_on Z & ( for x being Real st x in Z holds
(f1 `| Z) . x = 1 ) ) by A3, FDIFF_1:31;
A7: for x being Real st x in Z holds
f2 . x = ((- 1) * x) + a

proof

let x be Real; :: thesis: ( x in Z implies f2 . x = ((- 1) * x) + a )
assume x in Z ; :: thesis: f2 . x = ((- 1) * x) + a
then f2 . x = a - x by A1;
hence f2 . x = ((- 1) * x) + a ; :: thesis: verum

end;

then A8: ( f2 is_differentiable_on Z & ( for x being Real st x in Z holds
(f2 `| Z) . x = - 1 ) ) by A4, FDIFF_1:31;
A9: for x being Real st x in Z holds
f2 . x <> 0 by A1;
then A10: f1 / f2 is_differentiable_on Z by A6, A8, FDIFF_2:21;
for x being Real st x in Z holds
((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2 )

proof

let x be Real; :: thesis: ( x in Z implies ((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2 ) )
assume A11: x in Z ; :: thesis: ((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2 )
then A12: f1 is_differentiable_in x by A6, FDIFF_1:16;
A13: f2 is_differentiable_in x by A8, A11, FDIFF_1:16;
A14: ( f2 . x <> 0 & f1 . x = a + x & f2 . x = a - x ) by A1, A11;
then diff (f1 / f2),x = (((diff f1,x) * (f2 . x)) - ((diff f2,x) * (f1 . x))) / ((f2 . x) ^2 ) by A12, A13, FDIFF_2:14
.= ((((f1 `| Z) . x) * (f2 . x)) - ((diff f2,x) * (f1 . x))) / ((f2 . x) ^2 ) by A6, A11, FDIFF_1:def 8
.= ((((f1 `| Z) . x) * (f2 . x)) - (((f2 `| Z) . x) * (f1 . x))) / ((f2 . x) ^2 ) by A8, A11, FDIFF_1:def 8
.= ((1 * (f2 . x)) - (((f2 `| Z) . x) * (f1 . x))) / ((f2 . x) ^2 ) by A3, A5, A11, FDIFF_1:31
.= ((1 * (f2 . x)) - ((- 1) * (f1 . x))) / ((f2 . x) ^2 ) by A4, A7, A11, FDIFF_1:31
.= (2 * a) / ((a - x) ^2 ) by A14 ;
hence ((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2 ) by A10, A11, FDIFF_1:def 8; :: thesis: verum

end;

hence ( f1 / f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2 ) ) ) by A6, A8, A9, FDIFF_2:21; :: thesis: verum