reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem Th25:
for n be Nat, r be Real, f be PartFunc of S,T
 st 1 <= n & f is_differentiable_on n,Z
  holds r(#)f is_differentiable_on n,Z
proof
   let n be Nat, r be Real, f be PartFunc of S,T;
   assume A1: 1 <= n & f is_differentiable_on n,Z; then
A2:Z c= dom (r(#)f) by VFUNCT_1:def 4;
A3: Z is open by Th18,A1;
   for i be Nat st i <= n-1 holds diff(r(#)f,i,Z) is_differentiable_on Z
   proof
    let i be Nat;
    assume A4: i <= n-1;
    set H = diff_SP(i,S,T);
    set f1 = diff(f,i,Z);
A5: diff(f,i,Z) is_differentiable_on Z by A1,A4,Th14;
    n-1 - 0 <= n-1 + 1 by XREAL_1:7; then
A6: i <= n by A4,XXREAL_0:2; then
A7: diff(r(#)f,i,Z) = r(#)diff(f,i,Z) by A1,Th24;
    dom diff(f,i,Z) = Z by Th19,A6,A1; then
    Z c= dom (r(#)f1) by VFUNCT_1:def 4;
    hence diff(r(#)f,i,Z) is_differentiable_on Z by A3,A5,A7,NDIFF_1:41;
   end;
   hence thesis by Th14,A2;
end;
