reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T,
  x for set;

theorem Th29: :: Theorem 1.3.4. (a)
  for T being TopSpace, A being Subset of T holds Cl A = A \/ Der A
proof
  let T be TopSpace, A be Subset of T;
  per cases;
  suppose
A1: T is non empty;
    then
A2: Der A c= Cl A by Th28;
    thus Cl A c= A \/ Der A
    proof
      let x be object;
      assume
A3:   x in Cl A;
      then reconsider x9 = x as Point of T;
      per cases;
      suppose
        x in A;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose
A4:     not x in A;
        for U being open Subset of T st x9 in U ex y being Point of T st y
        in A /\ U & x9 <> y
        proof
          let U be open Subset of T;
          assume x9 in U;
          then A meets U by A3,PRE_TOPC:24;
          then consider y being object such that
A5:       y in A and
A6:       y in U by XBOOLE_0:3;
          reconsider y as Point of T by A5;
          take y;
          thus thesis by A4,A5,A6,XBOOLE_0:def 4;
        end;
        then x9 in Der A by A1,Th17;
        hence thesis by XBOOLE_0:def 3;
      end;
    end;
    let x be object;
    assume
A7: x in A \/ Der A;
    per cases by A7,XBOOLE_0:def 3;
    suppose
A8:   x in A;
      A c= Cl A by PRE_TOPC:18;
      hence thesis by A8;
    end;
    suppose
      x in Der A;
      hence thesis by A2;
    end;
  end;
  suppose
A9: T is empty;
    then the carrier of T is empty;
    then Cl A = {} \/ {} .= A \/ Der A by A9;
    hence thesis;
  end;
end;
