reserve X for non empty TopSpace,
  D for Subset of X;

theorem Th10:
  for X being anti-discrete non empty TopSpace for A being Subset
  of X holds (A <> the carrier of X implies Int A = {}) & (A = the carrier of X
  implies Int A = the carrier of X)
proof
  let X be anti-discrete non empty TopSpace;
  let A be Subset of X;
  thus A <> the carrier of X implies Int A = {}
  proof
    assume
A1: A <> the carrier of X;
    now
      assume Int A = the carrier of X;
      then the carrier of X c= A by TOPS_1:16;
      hence contradiction by A1,XBOOLE_0:def 10;
    end;
    hence thesis by TDLAT_3:18;
  end;
  thus A = the carrier of X implies Int A = the carrier of X
  proof
    assume A = the carrier of X;
    then A = [#]X;
    hence thesis by TOPS_1:15;
  end;
end;
