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 Th15:
  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: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A2: for x holds bdif(f1+f2,h).(k+1).x = bdif(f1,h).(k+1).x + bdif(f2,h
    ).(k+1).x;
    let x;
A3: 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 A2;
A4: bdif(f1+f2,h).(k+1) is Function of REAL,REAL by Th12;
A5: bdif(f2,h).(k+1) is Function of REAL,REAL by Th12;
A6: 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 A4,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 A3
      .= bD(bdif(f1,h).(k+1),h).x + (bdif(f2,h).(k+1).x - bdif(f2,h).(k+1).(
    x-h)) by A6,Th4
      .= bD(bdif(f1,h).(k+1),h).x + bD(bdif(f2,h).(k+1),h).x by A5,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;
A7: X[0]
  proof
    let x;
 reconsider xx=x, h as Element of REAL by XREAL_0:def 1;
    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.xx + f2.xx - (f1+f2).(xx-h) by VALUED_1:1
      .= f1.x + f2.x - (f1.(x-h) + f2.(x-h)) by VALUED_1:1
      .= (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;
  for n holds X[n] from NAT_1:sch 2(A7,A1);
  hence thesis;
end;
