theorem
  chi(A,X) is real-valued
proof
 for x st x in dom chi(A,X) holds |.chi(A,X).x.| < +infty
  proof
    let x;
    assume x in dom chi(A,X);
    per cases;
      suppose
     x in A;
then      chi(A,X).x = 1. by FUNCT_3:def 3;
then      |.chi(A,X).x.| = jj by EXTREAL1:def 1;
        hence thesis by XXREAL_0:9;
      end;
      suppose
     not x in A;
then      chi(A,X).x = 0. by FUNCT_3:def 3;
        hence thesis by EXTREAL1:def 1;
      end;
  end;
  hence thesis;
end;
