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
  for x0 being Real,f be PartFunc of REAL,REAL n,
  g be PartFunc of REAL,REAL-NS n, N being Neighbourhood of x0
st f=g & f is_differentiable_in x0 & N c= dom f
for h,c st rng c = {x0} & rng (h+c)c= N
holds h"(#)((g/*(h+c)) - g/*c) is convergent
& diff(f,x0) = lim (h"(#)((g/*(h+c)) - g/*c))
proof
  let x0 being Real,
      f be PartFunc of REAL,REAL n,
      g be PartFunc of REAL,REAL-NS n,
      N being Neighbourhood of x0;
  assume that
A1: f=g and
A2: f is_differentiable_in x0 and
A3: N c= dom f;
A4: g is_differentiable_in x0 by A1,A2;
    let h,c;
    assume rng c = {x0} & rng (h+c)c= N;
    then h"(#)((g/*(h+c)) - g/*c) is convergent
    & diff(g,x0) = lim (h"(#)((g/*(h+c)) - g/*c)) by A4,A1,A3,NDIFF_3:13;
    hence thesis by A1,Th3;
end;
