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 Th15:
  X misses Y implies (for x being Element of X holds (chi(Y,X)).x = 0)
  proof
    assume
A1: X misses Y;
    let x be Element of X;
    per cases;
    suppose A2: X is non empty;
      then not x in Y by A1,XBOOLE_0:def 4;
      then x in X \ Y by A2,XBOOLE_0:def 5;
      hence thesis by FUNCT_3:37;
    end;
    suppose X is empty;
      hence thesis;
    end;
  end;
