reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve y,X for set;
reserve x,x0,x1,x2,g,g1,g2,r,r1,s,p,p1,p2 for Real;
reserve i,m,k for Element of NAT;
reserve n,k for non zero Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq,seq1 for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve g,g1,g2 for PartFunc of REAL,REAL-NS n;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;

theorem Th13:
  Z c= dom 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 f and
A2: f is_differentiable_on Z;
  reconsider g=f as PartFunc of REAL,REAL-NS n by REAL_NS1:def 4;
A3: dom f = dom (r(#)f) by VALUED_2:def 39;
  reconsider r as Real;
A4: r(#)f = r(#)g by NFCONT_4:6;
A5: Z c= dom (r(#)g) by A3,A1,NFCONT_4:6;
A6: Z c= dom g by A2;
  now
    let x;
    assume x in Z;
    then f|Z is_differentiable_in x by A2;
    hence g|Z is_differentiable_in x;
  end;
  then g is_differentiable_on Z by A6,NDIFF_3:def 5;
   then
A7: r(#)g is_differentiable_on Z &
   for x st x in Z holds ((r(#)g)`|Z).x =r*diff(g,x) by A5,NDIFF_3:19;
A8: now
    let x;
    assume x in Z;
    then (r(#)g) |Z is_differentiable_in x by A7,NDIFF_3:def 5;
    hence (r(#)f) |Z is_differentiable_in x by A4;
  end;
  then
A9: r(#)f is_differentiable_on Z by A3,A1;
  now
    let x;
    assume A10: x in Z;
    then f is_differentiable_in x by A2,Th5;
    then r(#)f is_differentiable_in x & diff(r(#)f,x) = r*diff(f,x) by Th9;
    hence ((r(#)f)`|Z).x = r*diff(f,x) by A10,A9,Def4;
  end;
  hence thesis by A3,A8,A1;
end;
