reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem
for X be Subset of REAL m, f,g be PartFunc of REAL m,REAL n st
 f is_differentiable_on X & g is_differentiable_on X
 holds
    f-g is_differentiable_on X
  & for x be Element of REAL m st x in X
      holds ((f-g)`|X)/.x = diff(f,x)- diff(g,x)
proof
   let X be Subset of REAL m, f,g be PartFunc of REAL m,REAL n;
   assume
A1:f is_differentiable_on X & g is_differentiable_on X; then
A2: X is open by Th13; then
A3:X c= dom f & X c= dom g by A1,Th14;
   dom (f-g) = dom f /\ dom g by VALUED_2:def 46; then
A4:X c= dom (f-g) by A3,XBOOLE_1:19;
   now let x be Element of REAL m;
    assume x in X; then
    f is_differentiable_in x & g is_differentiable_in x by A1,A2,Th14;
    hence f-g is_differentiable_in x by PDIFF_6:21;
   end;
   hence f-g is_differentiable_on X by A4,A2,Th14;
   let x be Element of REAL m;
   assume A5: x in X; then
   f is_differentiable_in x & g is_differentiable_in x
   by A1,A2,Th14; then
   diff(f-g,x) = diff(f,x)- diff(g,x) by PDIFF_6:21;
   hence ((f-g)`|X)/.x = diff(f,x)- diff(g,x) by A4,A5,Def1;
end;
