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(r1(#)f1-r2(#)f2,h).(n+1).x =
  r1* fdif(f1,h).(n+1).x - r2* fdif(f2,h).(n+1).x
proof
  set g1=r1(#)f1;
  set g2=r2(#)f2;
  fdif(r1(#)f1-r2(#)f2,h).(n+1).x = fdif(g1,h).(n+1).x
       - fdif(g2,h).(n+1).x by DIFF_1:9
    .= r1* fdif(f1,h).(n+1).x - fdif(g2,h).(n+1).x by DIFF_1:7
    .= r1* fdif(f1,h).(n+1).x - r2* fdif(f2,h).(n+1).x by DIFF_1:7;
  hence thesis;
end;
