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;
