reserve n,m,k,i for Nat,
  h,r,r1,r2,x0,x1,x2,x for Real,
  S for Functional_Sequence of REAL,REAL,
  y for set;
reserve f,f1,f2 for Function of REAL,REAL;

theorem
  (fdif(fdif(f,h).m,h).n).x = fdif(f,h).(m+n).x
proof
  defpred X[Nat] means
for x holds (fdif(fdif(f,h).m,h).$1).x =
  fdif(f,h).(m+$1).x;
A1: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A2: for x holds (fdif(fdif(f,h).m,h).k).x = fdif(f,h).(m+k).x;
    let x;
A3: fdif(f,h).(m+k) is Function of REAL,REAL by Th2;
    fdif(f,h).m is Function of REAL,REAL by Th2;
    then
A4: fdif(fdif(f,h).m,h).k is Function of REAL,REAL by Th2;
    fdif(fdif(f,h).m,h).(k+1).x = fD(fdif(fdif(f,h).m,h).k,h).x by Def6
      .= fdif(fdif(f,h).m,h).k.(x+h) - fdif(fdif(f,h).m,h).k.x by A4,Th3
      .= fdif(f,h).(m+k).(x+h) - fdif(fdif(f,h).m,h).k.x by A2
      .= fdif(f,h).(m+k).(x+h) - fdif(f,h).(m+k).x by A2
      .= fD(fdif(f,h).(m+k),h).x by A3,Th3
      .= fdif(f,h).(m+k+1).x by Def6;
    hence thesis;
  end;
A5: X[0] by Def6;
  for n holds X[n] from NAT_1:sch 2(A5,A1);
  hence thesis;
end;
