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

theorem Th35:
  for A being Subset of X holds A is discrete iff for D being
  Subset of X st D c= A holds A /\ Cl D = D
proof
  let A be Subset of X;
  thus A is discrete implies for D being Subset of X st D c= A holds A /\ Cl D
  = D
  proof
    assume
A1: A is discrete;
    let D be Subset of X;
    assume
A2: D c= A;
    then consider F being Subset of X such that
A3: F is closed and
A4: A /\ F = D by A1;
    Cl D c= F by A3,A4,TOPS_1:5,XBOOLE_1:17;
    then
A5: A /\ Cl D c= D by A4,XBOOLE_1:26;
    D c= Cl D by PRE_TOPC:18;
    then D c= A /\ Cl D by A2,XBOOLE_1:19;
    hence thesis by A5;
  end;
  assume
A6: for D being Subset of X st D c= A holds A /\ Cl D = D;
  now
    let D be Subset of X;
    assume
A7: D c= A;
    now
      take F = Cl D;
      thus F is closed;
      thus A /\ F = D by A6,A7;
    end;
    hence ex F being Subset of X st F is closed & A /\ F = D;
  end;
  hence thesis;
end;
