reserve x,r,a,x0,p for Real;
reserve n,i,m for Element of NAT;
reserve Z for open Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;
reserve k for Nat;

theorem Th22:
  f is_differentiable_on n,Z implies r(#)f is_differentiable_on n, Z
proof
  assume
A1: f is_differentiable_on n,Z;
  now
    let i be Nat such that
A2: i <= n-1;
A3: i in NAT by ORDINAL1:def 12;
     diff(f,Z).i is_differentiable_on Z by A1,A2;
    then
A4: r(#) diff(f,Z).i is_differentiable_on Z by FDIFF_2:19;
    i <= n by A2,WSIERP_1:18;
    hence diff(r(#)f,Z).i is_differentiable_on Z by A1,A4,Th21,TAYLOR_1:23,A3;
  end;
  hence thesis;
end;
