reserve n,m for Element of NAT;
reserve h,k,r,r1,r2,x,x0,x1,x2,x3 for Real;
reserve f,f1,f2 for Function of REAL,REAL;

theorem
  (fdif(f,h).n).x = (cdif(f,h).n).(x+(n/2)*h)
  implies (bdif(f,h).n).x = (cdif(f,h).n).(x-(n/2)*h)
proof
  defpred X[Nat] means for x holds
  (bdif(f,h).$1).x = (cdif(f,h).$1).(x-($1/2)*h);
A1:X[0]
  proof
    let x;
    (bdif(f,h).0).x = f.x by DIFF_1:def 7
       .= (cdif(f,h).0).(x-(0/2)*h) by DIFF_1:def 8;
    hence thesis;
  end;
A2:for i be Nat st X[i] holds X[i+1]
  proof
    let i be Nat;
    assume
  A3:for x holds (bdif(f,h).i).x = (cdif(f,h).i).(x-(i/2)*h);
    let x;
  A4:bdif(f,h).i is Function of REAL,REAL by DIFF_1:12;
  A5:cdif(f,h).i is Function of REAL,REAL by DIFF_1:19;
    (bdif(f,h).(i+1)).x = bD(bdif(f,h).i,h).x by DIFF_1:def 7
      .= (bdif(f,h).i).x - (bdif(f,h).i).(x-h) by A4,DIFF_1:4
      .= (cdif(f,h).i).(x-(i/2)*h) - (bdif(f,h).i).(x-h) by A3
      .= (cdif(f,h).i).(x-(i/2)*h) - (cdif(f,h).i).((x-h)-(i/2)*h) by A3
      .= (cdif(f,h).i).((x-((i+1)/2)*h)+h/2)
         - (cdif(f,h).i).((x-((i+1)/2)*h)-h/2)
      .= cD(cdif(f,h).i,h).(x-((i+1)/2)*h) by A5,DIFF_1:5
      .= (cdif(f,h).(i+1)).(x-((i+1)/2)*h) by DIFF_1:def 8;
    hence thesis;
  end;
  for n be Nat holds X[n] from NAT_1:sch 2(A1,A2);
  hence thesis;
end;
