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 be PartFunc of REAL m,REAL n, r be Real
 st f is_differentiable_on X
 holds
    r(#)f is_differentiable_on X
  & for x be Element of REAL m st x in X holds ((r(#)f)`|X)/.x = r(#)diff(f,x)
proof
   let X be Subset of REAL m, f be PartFunc of REAL m,REAL n, r be Real;
   assume A1: f is_differentiable_on X; then
A2: X is open by Th13; then
   X c=dom f by A1,Th14; then
A3:X c= dom (r(#)f) by VALUED_2:def 39;
   now let x be Element of REAL m;
    assume x in X; then
    f is_differentiable_in x by A1,A2,Th14;
    hence r(#)f is_differentiable_in x by PDIFF_6:22;
   end;
   hence r(#)f is_differentiable_on X by A3,A2,Th14;
   let x be Element of REAL m;
   assume A4:x in X; then
   f is_differentiable_in x by A1,A2,Th14; then
   diff(r(#)f,x) = r(#)diff(f,x) by PDIFF_6:22;
   hence ((r(#)f)`|X)/.x = r(#)diff(f,x) by A3,A4,Def1;
end;
