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

theorem Th24:
  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 Subset of Y holds
({d0} c= A implies A is closed) & (A is non empty & A is closed implies {d0} c=
  A)
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 Subset of Y holds ({d0} c= A implies A is closed) & (A is
  non empty & A is closed implies {d0} c= A) by TOPS_3:79,Th20;
  assume
A1: the carrier of Y = D;
  assume
A2: for A being Subset of Y holds ({d0} c= A implies A is closed) & (A
  is non empty & A is closed implies {d0} c= A);
  now
    let A be Subset of Y, C be Subset of STS(D,d0);
    assume
A3: A = C;
A4: now
      assume
A5:   C is closed;
      now
        per cases;
        case
          C = {};
          hence A is closed by A3;
        end;
        case
          C <> {};
          then {d0} c= A by A3,A5,Th20;
          hence A is closed by A2;
        end;
      end;
      hence A is closed;
    end;
    now
      assume
A6:   A is closed;
      now
        per cases;
        case
          A = {};
          hence C is closed by A3;
        end;
        case
          A <> {};
          then {d0} c= C by A2,A3,A6;
          hence C is closed by Th20;
        end;
      end;
      hence C is closed;
    end;
    hence A is closed iff C is closed by A4;
  end;
  hence thesis by A1,TOPS_3:73;
end;
