theorem Th17:
  -- (A \ B) = (--A) \ (--B)
proof
  let z;
  hereby
    assume z in --(A\B);
    then
A1: -z in A \ B by Th12;
    then not -z in B by XBOOLE_0:def 5;
    then
A2: not z in --B by Th12;
    z in --A by A1,Th12;
    hence z in (--A) \ --B by A2,XBOOLE_0:def 5;
  end;
  assume
A3: z in (--A) \ --B;
  then not z in --B by XBOOLE_0:def 5;
  then
A4: not -z in B by Th12;
  -z in A by A3,Th12;
  then -z in A \ B by A4,XBOOLE_0:def 5;
  hence thesis by Th12;
end;
