let m be non zero Element of NAT ; :: thesis: for X being Subset of (REAL m)
for f, g being PartFunc of (REAL m),REAL st X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X holds
( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) )
let X be Subset of (REAL m); :: thesis: for f, g being PartFunc of (REAL m),REAL st X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X holds
( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) )
let f, g be PartFunc of (REAL m),REAL; :: thesis: ( X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X implies ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) ) )
assume A1: ( X c= dom f & X c= dom g ) ; :: thesis: ( not f is_differentiable_on X or not g is_differentiable_on X or ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) ) )
assume A2: ( f is_differentiable_on X & g is_differentiable_on X ) ; :: thesis: ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) )
then A3: X is open by A1, Th55;
dom (f - g) = (dom f) /\ (dom g) by VALUED_1:12;
then A4: X c= dom (f - g) by A1, XBOOLE_1:19;

A5: now :: thesis: for x being Element of REAL m st x in X holds
( f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) )

let x be Element of REAL m; :: thesis: ( x in X implies ( f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) ) )
assume x in X ; :: thesis: ( f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) )
then ( f is_differentiable_in x & g is_differentiable_in x ) by A1, A2, A3, Th54;
hence ( f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) ) by Th51; :: thesis: verum

end;

then for x being Element of REAL m st x in X holds
f - g is_differentiable_in x ;
hence f - g is_differentiable_on X by A4, A3, Th54; :: thesis: for x being Element of REAL m st x in X holds
((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x)
let x be Element of REAL m; :: thesis: ( x in X implies ((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) )
assume A6: x in X ; :: thesis: ((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x)
then ((f - g) `| X) /. x = diff ((f - g),x) by A4, Def4;
then ((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) by A6, A5;
then ((f - g) `| X) /. x = ((f `| X) /. x) - (diff (g,x)) by A1, A6, Def4;
hence ((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) by A1, A6, Def4; :: thesis: verum