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(f,h).(2*n)).x = (cdif(f,h).(2*n)).(x+n*h)
proof
  defpred X[Nat] means
  for x holds (fdif(f,h).(2*$1)).x = (cdif(f,h).(2*$1)).(x+$1*h);
A1: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A2: for x holds (fdif(f,h).(2*k)).x = cdif(f,h).(2*k).(x+k*h);
    let x;
A3: fdif(f,h).(2*k).(x+h+h) = cdif(f,h).(2*k).(x+h+h+k*h) by A2
      .= cdif(f,h).(2*k).(x+(k+2)*h);
A4: fdif(f,h).(2*k).(x+h) = cdif(f,h).(2*k).(x+h+k*h) by A2
      .= cdif(f,h).(2*k).(x+(k+1)*h);
    set r3 = cdif(f,h).(2*k).(x+k*h);
    set r2 = cdif(f,h).(2*k).(x+(k+1)*h);
    set r1 = cdif(f,h).(2*k).(x+(k+2)*h);
A5: fdif(f,h).(2*k+1) is Function of REAL,REAL by Th2;
A6: cdif(f,h).(2*k) is Function of REAL,REAL by Th19;
A7: cdif(f,h).(2*k+1) is Function of REAL,REAL by Th19;
A8: fdif(f,h).(2*k) is Function of REAL,REAL by Th2;
A9: cdif(f,h).(2*k+1).(x+(k+1)*h-h/2) = cD(cdif(f,h).(2*k),h).(x+(k+1)*h-
    h/2) by Def8
      .= cdif(f,h).(2*k).(x+(k+1)*h-h/2+h/2) - cdif(f,h).(2*k).(x+(k+1)*h-h/
    2-h/2) by A6,Th5
      .= cdif(f,h).(2*k).(x+(k+1)*h) - cdif(f,h).(2*k).(x+k*h);
A10: cdif(f,h).(2*k+1).(x+(k+1)*h+h/2) = cD(cdif(f,h).(2*k),h).(x+(k+1)*h+
    h/2) by Def8
      .= cdif(f,h).(2*k).(x+(k+1)*h+h/2+h/2) - cdif(f,h).(2*k).(x+(k+1)*h+h/
    2-h/2) by A6,Th5
      .= cdif(f,h).(2*k).(x+(k+2)*h) - cdif(f,h).(2*k).(x+(k+1)*h);
A11: cdif(f,h).(2*(k+1)).(x+(k+1)*h) = cdif(f,h).(2*k+1+1).(x+(k+1)*h)
      .= cD(cdif(f,h).(2*k+1),h).(x+(k+1)*h) by Def8
      .= (r1 - r2) - (r2 - r3) by A7,A10,A9,Th5;
    fdif(f,h).(2*(k+1)).x = fdif(f,h).(2*k+1+1).x
      .= fD(fdif(f,h).(2*k+1),h).x by Def6
      .= (fdif(f,h).(2*k+1)).(x+h) - (fdif(f,h).(2*k+1)).x by A5,Th3
      .= fD(fdif(f,h).(2*k),h).(x+h) - (fdif(f,h).(2*k+1)).x by Def6
      .= fD(fdif(f,h).(2*k),h).(x+h) - fD(fdif(f,h).(2*k),h).x by Def6
      .= fdif(f,h).(2*k).(x+h+h) - fdif(f,h).(2*k).(x+h) - fD(fdif(f,h).(2*k
    ),h).x by A8,Th3
      .= fdif(f,h).(2*k).(x+h+h) - fdif(f,h).(2*k).(x+h) - (fdif(f,h).(2*k).
    (x+h) - fdif(f,h).(2*k).x) by A8,Th3
      .= cdif(f,h).(2*k).(x+(k+2)*h) - cdif(f,h).(2*k).(x+(k+1)*h) - (cdif(f
    ,h).(2*k).(x+(k+1)*h) - cdif(f,h).(2*k).(x+k*h)) by A2,A3,A4;
    hence thesis by A11;
  end;
A12: X[0]
  proof
    let x;
    (fdif(f,h).(2*0)).x = f.x by Def6
      .= (cdif(f,h).(2*0)).(x+0*h) by Def8;
    hence thesis;
  end;
  for n holds X[n] from NAT_1:sch 2(A12,A1);
  hence thesis;
end;
