theorem Th16:
  -- (A /\ B) = (--A) /\ (--B)
proof
  let z;
  hereby
    assume z in --(A/\B);
    then
A1: -z in A /\ B by Th12;
    then -z in B by XBOOLE_0:def 4;
    then
A2: z in --B by Th12;
    -z in A by A1,XBOOLE_0:def 4;
    then z in --A by Th12;
    hence z in (--A) /\ --B by A2,XBOOLE_0:def 4;
  end;
  assume
A3: z in (--A) /\ --B;
  then z in --B by XBOOLE_0:def 4;
  then
A4: -z in B by Th12;
  z in --A by A3,XBOOLE_0:def 4;
  then -z in A by Th12;
  then -z in A /\ B by A4,XBOOLE_0:def 4;
  hence thesis by Th12;
end;
