reserve X for non empty TopSpace,
  D for Subset of X;
reserve D for non empty set,
  d0 for Element of D;

theorem
  for Y being non empty TopSpace holds the TopStruct of Y = the
TopStruct of STS(D,d0) iff the carrier of Y = D & for A being non empty Subset
  of Y holds Cl A = A \/ {d0}
proof
  let Y be non empty TopSpace;
  thus the TopStruct of Y = the TopStruct of STS(D,d0) implies the carrier of
  Y = D & for A being non empty Subset of Y holds Cl A = A \/ {d0}
  proof
    assume
A1: the TopStruct of Y = the TopStruct of STS(D,d0);
    hence the carrier of Y = D;
    reconsider P = {d0} as Subset of Y by A1;
    now
      let A be non empty Subset of Y;
      reconsider B = A as Subset of Y;
      Cl A is non empty by PCOMPS_1:2;
      then
A2:   {d0} c= Cl A by A1,Th24;
      A c= Cl A by PRE_TOPC:18;
      then
A3:   A \/ {d0} c= Cl A by A2,XBOOLE_1:8;
      {d0} c= A \/ P by XBOOLE_1:7;
      then B \/ P is closed by A1,Th24;
      then
A4:   Cl(A \/ P) = A \/ {d0} by PRE_TOPC:22;
      Cl A c= Cl(A \/ P) by PRE_TOPC:19,XBOOLE_1:7;
      hence Cl A = A \/ {d0} by A4,A3,XBOOLE_0:def 10;
    end;
    hence thesis;
  end;
  assume
A5: the carrier of Y = D;
  assume
A6: for A being non empty Subset of Y holds Cl A = A \/ {d0};
  now
    let A be Subset of Y;
    thus {d0} c= A implies A is closed
    proof
      assume {d0} c= A;
      then A = A \/ {d0} by XBOOLE_1:12;
      then Cl A = A by A6;
      hence thesis;
    end;
    thus A is non empty & A is closed implies {d0} c= A
    proof
      assume A is non empty;
      then
A7:   Cl A = A \/ {d0} by A6;
      assume A is closed;
      then
A8:   Cl A = A by PRE_TOPC:22;
      assume not {d0} c= A;
      hence contradiction by A7,A8,XBOOLE_1:7;
    end;
  end;
  hence thesis by A5,Th24;
end;
