theorem
  (for n holds F.n = Y /\ great_eq_dom(f,r+1/(n+1))) implies Y /\
  great_dom(f,r) = union rng F
proof
  assume for n holds F.n = Y /\ great_eq_dom(f,r+1/(n+1));
  then
  for n be Element of NAT holds F.n =Y /\ great_eq_dom(R_EAL f,(r+1/(
  n+1)));
  then Y /\ great_dom(f,r) = union rng F by MESFUNC1:22;
  hence thesis;
end;
