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(r(#)f1+f2,h).(n+1).x = r* bdif(f1,h).(n+1).x + bdif(f2,h).(n+1).x
proof
  set g=r(#)f1;
  bdif(r(#)f1+f2,h).(n+1).x = bdif(g,h).(n+1).x + bdif(f2,h).(n+1).x
                                                       by DIFF_1:15
    .= r* bdif(f1,h).(n+1).x + bdif(f2,h).(n+1).x by DIFF_1:14;
  hence thesis;
end;
