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 Th22:
  cdif(f1+f2,h).(n+1).x = cdif(f1,h).(n+1).x + cdif(f2,h).(n+1).x
proof
  defpred X[Nat] means
for x holds cdif(f1+f2,h).($1+1).x = cdif(f1
  ,h).($1+1).x + cdif(f2,h).($1+1).x;
A1: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A2: for x holds cdif(f1+f2,h).(k+1).x = cdif(f1,h).(k+1).x + cdif(f2,h
    ).(k+1).x;
    let x;
A3: cdif(f1+f2,h).(k+1).(x-h/2) = cdif(f1,h).(k+1).(x-h/2) + cdif(f2,h).(k
+1).(x- h/2) & cdif(f1+f2,h).(k+1).(x+h/2) = cdif(f1,h).(k+1).(x+h/2) + cdif(f2
    ,h).(k+ 1).(x+h/2) by A2;
A4: cdif(f1+f2,h).(k+1) is Function of REAL,REAL by Th19;
A5: cdif(f2,h).(k+1) is Function of REAL,REAL by Th19;
A6: cdif(f1,h).(k+1) is Function of REAL,REAL by Th19;
    cdif(f1+f2,h).(k+1+1).x = cD(cdif(f1+f2,h).(k+1),h).x by Def8
      .= cdif(f1+f2,h).(k+1).(x+h/2) - cdif(f1+f2,h).(k+1).(x-h/2) by A4,Th5
      .= (cdif(f1,h).(k+1).(x+h/2) - cdif(f1,h).(k+1).(x-h/2)) + (cdif(f2,h)
    .(k+1).(x+h/2) - cdif(f2,h).(k+1).(x-h/2)) by A3
      .= cD(cdif(f1,h).(k+1),h).x + (cdif(f2,h).(k+1).(x+h/2) - cdif(f2,h).(
    k+1).(x-h/2)) by A6,Th5
      .= cD(cdif(f1,h).(k+1),h).x + cD(cdif(f2,h).(k+1),h).x by A5,Th5
      .= cdif(f1,h).(k+1+1).x + cD(cdif(f2,h).(k+1),h).x by Def8
      .= cdif(f1,h).(k+1+1).x + cdif(f2,h).(k+1+1).x by Def8;
    hence thesis;
  end;
A7: X[0]
  proof
    let x;
 reconsider xx=x, hp = h/2 as Element of REAL by XREAL_0:def 1;
    cdif(f1+f2,h).(0+1).x = cD(cdif(f1+f2,h).0,h).x by Def8
      .= cD(f1+f2,h).x by Def8
      .= (f1+f2).(x+h/2) - (f1+f2).(x-h/2) by Th5
      .= f1.(xx+h/2) + f2.(xx+hp) - (f1+f2).(xx-hp) by VALUED_1:1
      .= f1.(x+h/2) + f2.(x+hp) - (f1.(x-h/2) + f2.(x-hp)) by VALUED_1:1
      .= (f1.(x+h/2) - f1.(x-h/2)) + (f2.(x+h/2) - f2.(x-h/2))
      .= cD(f1,h).x + (f2.(x+h/2) - f2.(x-h/2)) by Th5
      .= cD(f1,h).x + cD(f2,h).x by Th5
      .= cD(cdif(f1,h).0,h).x + cD(f2,h).x by Def8
      .= cD(cdif(f1,h).0,h).x + cD(cdif(f2,h).0,h).x by Def8
      .= cdif(f1,h).(0+1).x + cD(cdif(f2,h).0,h).x by Def8
      .= cdif(f1,h).(0+1).x + cdif(f2,h).(0+1).x by Def8;
    hence thesis;
  end;
  for n holds X[n] from NAT_1:sch 2(A7,A1);
  hence thesis;
end;
