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 = x - a & f2 . x = x + a & 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) / ((x + a) ^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 = x - a & f2 . x = x + a & 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) / ((x + a) ^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 = x - a & f2 . x = x + a & 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) / ((x + a) ^2) ) ) )
assume that
A1: Z c= dom (f1 / f2) and
A2: for x being Real st x in Z holds
( f1 . x = x - a & f2 . x = x + a & 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) / ((x + a) ^2) ) )
A3: for x being Real st x in Z holds
f2 . x = (1 * x) + a by A2;
A4: Z c= (dom f1) /\ ((dom f2) \ (f2 " {0})) by A1, RFUNCT_1:def 1;
then A5: Z c= dom f1 by XBOOLE_1:18;
A6: Z c= dom f2 by A4, XBOOLE_1:1;
then A7: f2 is_differentiable_on Z by A3, FDIFF_1:23;
A8: for x being Real st x in Z holds
f1 . x = (1 * x) + (- a)

proof

let x be Real; :: thesis: ( x in Z implies f1 . x = (1 * x) + (- a) )
A9: (1 * x) + (- a) = (1 * x) - a ;
assume x in Z ; :: thesis: f1 . x = (1 * x) + (- a)
hence f1 . x = (1 * x) + (- a) by A2, A9; :: thesis: verum

end;

then A10: f1 is_differentiable_on Z by A5, FDIFF_1:23;
A11: for x being Real st x in Z holds
f2 . x <> 0 by A2;
then A12: f1 / f2 is_differentiable_on Z by A10, A7, FDIFF_2:21;
for x being Real st x in Z holds
((f1 / f2) `| Z) . x = (2 * a) / ((x + a) ^2)

proof

let x be Real; :: thesis: ( x in Z implies ((f1 / f2) `| Z) . x = (2 * a) / ((x + a) ^2) )
assume A13: x in Z ; :: thesis: ((f1 / f2) `| Z) . x = (2 * a) / ((x + a) ^2)
then A14: f2 . x <> 0 by A2;
A15: ( f1 . x = x - a & f2 . x = x + a ) by A2, A13;
( f1 is_differentiable_in x & f2 is_differentiable_in x ) by A10, A7, A13, FDIFF_1:9;
then diff ((f1 / f2),x) = (((diff (f1,x)) * (f2 . x)) - ((diff (f2,x)) * (f1 . x))) / ((f2 . x) ^2) by A14, FDIFF_2:14
.= ((((f1 `| Z) . x) * (f2 . x)) - ((diff (f2,x)) * (f1 . x))) / ((f2 . x) ^2) by A10, A13, FDIFF_1:def 7
.= ((((f1 `| Z) . x) * (f2 . x)) - (((f2 `| Z) . x) * (f1 . x))) / ((f2 . x) ^2) by A7, A13, FDIFF_1:def 7
.= ((1 * (f2 . x)) - (((f2 `| Z) . x) * (f1 . x))) / ((f2 . x) ^2) by A5, A8, A13, FDIFF_1:23
.= ((1 * (f2 . x)) - (1 * (f1 . x))) / ((f2 . x) ^2) by A6, A3, A13, FDIFF_1:23
.= (2 * a) / ((x + a) ^2) by A15 ;
hence ((f1 / f2) `| Z) . x = (2 * a) / ((x + a) ^2) by A12, A13, FDIFF_1:def 7; :: 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) / ((x + a) ^2) ) ) by A10, A7, A11, FDIFF_2:21; :: thesis: verum