theorem
  Y c= X & f is_bounded_on X implies f is_bounded_on Y
proof
  assume that
A1: Y c= X and
A2: f is_bounded_on X;
  consider r such that
A3: for c st c in X /\ dom f holds ||.f/.c.|| <= r by A2;
  take r;
  let c;
  assume c in Y /\ dom f;
  then c in Y & c in dom f by XBOOLE_0:def 4;
  then c in X /\ dom f by A1,XBOOLE_0:def 4;
  hence thesis by A3;
end;
