theorem
  (for n holds F.n = Y /\ less_eq_dom(f,r-1/(n+1))) implies Y /\
  less_dom(f,r) = union rng F
proof
  assume for n holds F.n = Y /\ less_eq_dom(f,r-1/(n+1));
  then
  for n be Element of NAT holds F.n =Y /\ less_eq_dom(R_EAL f,(r-1/(n
  +1)));
  then Y /\ less_dom(f,r) = union rng F by MESFUNC1:21;
  hence thesis;
end;
