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;
