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(f1+r(#)f2,h).(n+1).x = fdif(f1,h).(n+1).x + r* fdif(f2,h).(n+1).x
proof
  set g=r(#)f2;
  fdif(f1+r(#)f2,h).(n+1).x = fdif(f1,h).(n+1).x + fdif(g,h).(n+1).x
                                                          by DIFF_1:8
    .= fdif(f1,h).(n+1).x + r* fdif(f2,h).(n+1).x by DIFF_1:7;
  hence thesis;
end;
