reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve X for set;
reserve x,x0,g,r,s,p for Real;
reserve n,m,k for Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,the carrier of F;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;
reserve R,R1,R2 for RestFunc of F;
reserve L,L1,L2 for LinearFunc of F;

theorem
  Z c= dom (r(#)f) & f is_differentiable_on Z implies
  r(#)f is_differentiable_on Z &
  for x st x in Z holds ((r(#) f)`|Z).x =r*diff(f,x)
  proof
    assume that
    A1: Z c= dom (r(#)f) and
    A2: f is_differentiable_on Z;
    now
      let x0;
      assume x0 in Z;
      then f is_differentiable_in x0 by A2,Th10;
      hence r(#)f is_differentiable_in x0 by Th16;
    end;
    hence
    A3: r(#)f is_differentiable_on Z by A1,Th10;
    let x;
    assume
    A4: x in Z; then
    A5: f is_differentiable_in x by A2,Th10;
    thus ((r(#)f)`|Z).x = diff((r(#)f),x) by A3,A4,Def6
     .= r*diff(f,x) by A5,Th16;
  end;
