reserve h,h1,h2 for 0-convergent non-zero Real_Sequence,
  c,c1 for constant Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x0,r,r0,r1,r2,g,g1,g2 for Real,
  n0,k,n,m for Element of NAT,
  a,b,d for Real_Sequence,
  x for set;

theorem Th15:
 for g1 being Real holds
  f is_right_differentiable_in x0 & Rdiff(f,x0) = g1 iff (ex r st
r>0 & [.x0,x0+r.] c= dom f) & for h,c st rng c = {x0} & rng (h+c) c= dom f & (
for n being Nat holds h.n > 0)
holds h"(#)(f/*(h+c) - f/*c) is convergent & lim (h"(#)(f
  /*(h+c) - f/*c)) = g1
proof let g1 be Real;
  thus f is_right_differentiable_in x0 & Rdiff(f,x0) = g1 implies (ex r st r>0
& [.x0,x0+r.] c= dom f) & for h,c st rng c = {x0} & rng (h+c) c= dom f &
 (for n being Nat
holds h.n > 0) holds h"(#)(f/*(h+c) - f/*c) is convergent & lim (h"(#)(f/*(h+c)
  - f/*c)) = g1 by Def6;
  assume that
A1: ex r st r>0 & [.x0,x0+r.] c= dom f and
A2: for h,c st rng c = {x0} & rng (h+c) c= dom f &
   (for n being Nat holds h.n > 0)
holds h"(#)(f/*(h+c) - f/*c) is convergent & lim (h"(#)(f/*(h+c) - f/*c)) = g1;
  for h,c holds ( rng c = {x0} & rng (h+c) c= dom f &
  for n being Nat holds h.n > 0
  ) implies h"(#)(f/*(h+c) - f/*c) is convergent by A2;
  hence
A3: f is_right_differentiable_in x0 by A1;
  for h,c holds ( rng c = {x0} & rng (h+c) c= dom f &
   for n being Nat holds h.n > 0)
  implies lim (h"(#)(f/*(h+c) - f/*c)) = g1 by A2;
  hence thesis by A3,Def6;
end;
