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 Th8:
  fdif(f1+f2,h).(n+1).x = fdif(f1,h).(n+1).x + fdif(f2,h).(n+1).x
proof
  defpred X[Nat] means
for x holds fdif(f1+f2,h).($1+1).x = fdif(f1
  ,h).($1+1).x + fdif(f2,h).($1+1).x;
A1: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A2: for x holds fdif(f1+f2,h).(k+1).x = fdif(f1,h).(k+1).x + fdif(f2,h
    ).(k+1).x;
    let x;
A3: fdif(f1+f2,h).(k+1).x = fdif(f1,h).(k+1).x + fdif(f2,h).(k+1).x & fdif
(f1+f2,h).(k+1).(x+h) = fdif(f1,h).(k+1).(x+h) + fdif(f2,h).(k+1).(x+h) by A2;
A4: fdif(f1+f2,h).(k+1) is Function of REAL,REAL by Th2;
A5: fdif(f2,h).(k+1) is Function of REAL,REAL by Th2;
A6: fdif(f1,h).(k+1) is Function of REAL,REAL by Th2;
    fdif(f1+f2,h).(k+1+1).x = fD(fdif(f1+f2,h).(k+1),h).x by Def6
      .= fdif(f1+f2,h).(k+1).(x+h) - fdif(f1+f2,h).(k+1).x by A4,Th3
      .= (fdif(f1,h).(k+1).(x+h) - fdif(f1,h).(k+1).x) + (fdif(f2,h).(k+1).(
    x+h) - fdif(f2,h).(k+1).x) by A3
      .= fD(fdif(f1,h).(k+1),h).x + (fdif(f2,h).(k+1).(x+h) - fdif(f2,h).(k+
    1).x) by A6,Th3
      .= fD(fdif(f1,h).(k+1),h).x + fD(fdif(f2,h).(k+1),h).x by A5,Th3
      .= fdif(f1,h).(k+1+1).x + fD(fdif(f2,h).(k+1),h).x by Def6
      .= fdif(f1,h).(k+1+1).x + fdif(f2,h).(k+1+1).x by Def6;
    hence thesis;
  end;
A7: X[0]
  proof
    let x;
 reconsider xx=x, h as Element of REAL by XREAL_0:def 1;
    fdif(f1+f2,h).(0+1).x = fD(fdif(f1+f2,h).0,h).x by Def6
      .= fD(f1+f2,h).x by Def6
      .= (f1+f2).(x+h) - (f1+f2).x by Th3
      .= f1.(xx+h) + f2.(xx+h) - (f1+f2).xx by VALUED_1:1
      .= f1.(x+h) + f2.(x+h) - (f1.x + f2.x) by VALUED_1:1
      .= (f1.(x+h) - f1.x) + (f2.(x+h) - f2.x)
      .= fD(f1,h).x + (f2.(x+h) - f2.x) by Th3
      .= fD(f1,h).x + fD(f2,h).x by Th3
      .= fD(fdif(f1,h).0,h).x + fD(f2,h).x by Def6
      .= fD(fdif(f1,h).0,h).x + fD(fdif(f2,h).0,h).x by Def6
      .= fdif(f1,h).(0+1).x + fD(fdif(f2,h).0,h).x by Def6
      .= fdif(f1,h).(0+1).x + fdif(f2,h).(0+1).x by Def6;
    hence thesis;
  end;
  for n holds X[n] from NAT_1:sch 2(A7,A1);
  hence thesis;
end;
