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
  f is_differentiable_in x0 implies f is_right_differentiable_in x0 & f
is_left_differentiable_in x0 & diff(f,x0) = Rdiff(f,x0) & diff(f,x0) = Ldiff(f,
  x0)
proof
  assume
A1: f is_differentiable_in x0;
  then consider N be Neighbourhood of x0 such that
A2: N c= dom f by FDIFF_2:11;
  consider g1 be Real such that
A3: g1 > 0 and
A4: N = ].x0 - g1,x0 + g1.[ by RCOMP_1:def 6;
A5: g1 > g1/2 by A3,XREAL_1:216;
A6: ex r st r>0 & [.x0,x0+r.] c= dom f
  proof
    take r = g1/2;
    thus r > 0 by A3,XREAL_1:215;
    |.(x0 + g1/2) -x0 .| = g1/2 by A3,ABSVALUE:def 1;
    then
A7: x0 + r in ].x0 -g1,x0 +g1.[ by A5,RCOMP_1:1;
    x0 in ].x0 - g1, x0 + g1 .[ by A4,RCOMP_1:16;
    then [.x0,x0 +r.] c= ].x0 -g1,x0 +g1.[ by A7,XXREAL_2:def 12;
    hence thesis by A2,A4;
  end;
A8: ex r st 0 < r & [.x0 -r,x0.] c= dom f
  proof
    take r = g1/2;
    thus r > 0 by A3,XREAL_1:215;
    |.(x0 - g1/2) -x0 .| =|.-g1/2 + 0 .| .=|.g1/2 .|by COMPLEX1:52
      .=g1/2 by A3,ABSVALUE:def 1;
    then
A9: x0 - r in ].x0 -g1,x0 +g1.[ by A5,RCOMP_1:1;
    x0 in ].x0 - g1, x0 + g1 .[ by A4,RCOMP_1:16;
    then [.x0 -r,x0 .] c= ].x0 -g1,x0 +g1.[ by A9,XXREAL_2:def 12;
    hence thesis by A2,A4;
  end;
  diff(f,x0) = diff(f,x0);
  then ( 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)) =
diff(f,x0) )& 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)) =
  diff(f,x0) by A1,FDIFF_2:12;
  hence thesis by A6,A8,Th9,Th15;
end;
