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
  bdif(f1-f2,h).(n+1).x = bdif(f1,h).(n+1).x - bdif(f2,h).(n+1).x
proof
  defpred X[Nat] means
for x holds bdif(f1-f2,h).($1+1).x = bdif(f1
  ,h).($1+1).x - bdif(f2,h).($1+1).x;
A1: X[0]
  proof
    let x;
    x in REAL by XREAL_0:def 1;
    then x in dom f1 & x in dom f2 by FUNCT_2:def 1;
    then x in dom f1 /\ dom f2 by XBOOLE_0:def 4;
    then
A2: x in dom (f1-f2) by VALUED_1:12;
    x-h in REAL by XREAL_0:def 1;
    then x-h in dom f1 & x-h in dom f2 by FUNCT_2:def 1;
    then x-h in dom f1 /\ dom f2 by XBOOLE_0:def 4;
    then
A3: x-h in dom (f1-f2) by VALUED_1:12;
    bdif(f1-f2,h).(0+1).x = bD(bdif(f1-f2,h).0,h).x by Def7
      .= bD(f1-f2,h).x by Def7
      .= (f1-f2).x - (f1-f2).(x-h) by Th4
      .= f1.x - f2.x - (f1-f2).(x-h) by A2,VALUED_1:13
      .= f1.x - f2.x - (f1.(x-h) - f2.(x-h)) by A3,VALUED_1:13
      .= (f1.x - f1.(x-h)) - (f2.x - f2.(x-h))
      .= bD(f1,h).x - (f2.x - f2.(x-h)) by Th4
      .= bD(f1,h).x - bD(f2,h).x by Th4
      .= bD(bdif(f1,h).0,h).x - bD(f2,h).x by Def7
      .= bD(bdif(f1,h).0,h).x - bD(bdif(f2,h).0,h).x by Def7
      .= bdif(f1,h).(0+1).x - bD(bdif(f2,h).0,h).x by Def7
      .= bdif(f1,h).(0+1).x - bdif(f2,h).(0+1).x by Def7;
    hence thesis;
  end;
A4: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A5: for x holds bdif(f1-f2,h).(k+1).x = bdif(f1,h).(k+1).x - bdif(f2,
    h).(k+1).x;
    let x;
A6: bdif(f1-f2,h).(k+1).x = bdif(f1,h).(k+1).x - bdif(f2,h).(k+1).x &
bdif(f1-f2,h).(k+1).(x-h) = bdif(f1,h).(k+1).(x-h) - bdif(f2,h).(k+1).(x-h) by
A5;
A7: bdif(f1-f2,h).(k+1) is Function of REAL,REAL by Th12;
A8: bdif(f2,h).(k+1) is Function of REAL,REAL by Th12;
A9: bdif(f1,h).(k+1) is Function of REAL,REAL by Th12;
    bdif(f1-f2,h).(k+1+1).x = bD(bdif(f1-f2,h).(k+1),h).x by Def7
      .= bdif(f1-f2,h).(k+1).x - bdif(f1-f2,h).(k+1).(x-h) by A7,Th4
      .= (bdif(f1,h).(k+1).x - bdif(f1,h).(k+1).(x-h)) - (bdif(f2,h).(k+1).x
    - bdif(f2,h).(k+1).(x-h)) by A6
      .= bD(bdif(f1,h).(k+1),h).x - (bdif(f2,h).(k+1).x - bdif(f2,h).(k+1).(
    x-h)) by A9,Th4
      .= bD(bdif(f1,h).(k+1),h).x - bD(bdif(f2,h).(k+1),h).x by A8,Th4
      .= bdif(f1,h).(k+1+1).x - bD(bdif(f2,h).(k+1),h).x by Def7
      .= bdif(f1,h).(k+1+1).x - bdif(f2,h).(k+1+1).x by Def7;
    hence thesis;
  end;
  for n holds X[n] from NAT_1:sch 2(A1,A4);
  hence thesis;
end;
