reserve a,b,c,d,e for Real;
reserve X,Y for set,
          Z for non empty set,
          r for Real,
          s for ExtReal,
          A for Subset of REAL,
          f for real-valued Function;

theorem Th17:
  chi(X,X) is positive-yielding
  proof
    now
      let r be Real;
      assume
A1:   r in rng chi(X,X);
      r <> 0
      proof
        assume
A2:     r = 0;
        ex s be Element of X st s in dom chi(X,X) & r = (chi(X,X)).s
          by A1,PARTFUN1:3;
        hence contradiction by A2,FUNCT_3:def 3;
      end;
      hence 0 < r by A1;
    end;
    hence thesis by PARTFUN3:def 1;
  end;
