theorem Th15:
  -- (A \/ B) = (--A) \/ (--B)
proof
  let z;
  hereby
    assume z in --(A\/B);
    then -z in A \/ B by Th12;
    then -z in A or -z in B by XBOOLE_0:def 3;
    then z in --A or z in --B by Th12;
    hence z in --A \/ --B by XBOOLE_0:def 3;
  end;
  assume z in --A \/ --B;
  then z in --A or z in --B by XBOOLE_0:def 3;
  then -z in A or -z in B by Th12;
  then -z in A \/ B by XBOOLE_0:def 3;
  hence thesis by Th12;
end;
