reserve Y for TopStruct;
reserve X for non empty TopSpace;
reserve X for almost_discrete non empty TopSpace;

theorem Th49:
  for a, b being Point of X holds a in Cl {b} implies Cl {a} = Cl {b}
proof
  let a, b be Point of X;
  assume a in Cl {b};
  then
A1: {a} c= Cl {b} by ZFMISC_1:31;
  b in Cl {a}
  proof
    assume not b in Cl {a};
    then
A2: {b} misses Cl {a} by ZFMISC_1:50;
    Cl {a} is open by TDLAT_3:22;
    then Cl {b} misses Cl {a} by A2,TSEP_1:36;
    then Cl {b} /\ Cl {a} = {};
    then Cl {a} = {} by A1,TOPS_1:5,XBOOLE_1:28;
    hence contradiction by PRE_TOPC:18,XBOOLE_1:3;
  end;
  then {b} c= Cl {a} by ZFMISC_1:31;
  then
A3: Cl {b} c= Cl {a} by TOPS_1:5;
  Cl {a} c= Cl {b} by A1,TOPS_1:5;
  hence thesis by A3;
end;
