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
  bdif(r1(#)f1-r2(#)f2,h).(n+1).x =
  r1* bdif(f1,h).(n+1).x - r2* bdif(f2,h).(n+1).x
proof
  set g1=r1(#)f1;
  set g2=r2(#)f2;
  bdif(r1(#)f1-r2(#)f2,h).(n+1).x = bdif(g1,h).(n+1).x
       - bdif(g2,h).(n+1).x by DIFF_1:16
    .= r1* bdif(f1,h).(n+1).x - bdif(g2,h).(n+1).x by DIFF_1:14
    .= r1* bdif(f1,h).(n+1).x - r2* bdif(f2,h).(n+1).x by DIFF_1:14;
  hence thesis;
end;
