theorem Th5:
  for r holds ex f being Function of A,REAL st rng f = {r} & f|A is
  bounded
proof
  let r;
  r(#)chi(A,A) is total by Th3;
  then reconsider f=r(#)chi(A,A) as Function of A,REAL;
A1: rng f = {r} by Th3;
  then
A2: f|A is bounded_below by INTEGRA1:12;
  f|A is bounded_above by A1,INTEGRA1:14;
  hence thesis by A1,A2;
end;
