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 Subset of Y st A
  <> D holds Int 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 Subset of Y st A <> D holds Int 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 Subset of Y;
      reconsider B = A as Subset of Y;
A2:   A = A /\ D by A1,XBOOLE_1:28;
      assume
A3:   A <> D;
      now
        assume Int A = D;
        then D c= A by TOPS_1:16;
        hence contradiction by A1,A3,XBOOLE_0:def 10;
      end;
      then
A4:   Int A c= D \ P by A1,Th25;
      A \ {d0} c= D \ {d0} by A1,XBOOLE_1:33;
      then B \ P is open by A1,Th25;
      then Int(A \ P) = A \ P by TOPS_1:23;
      then
A5:   A \ {d0} c= Int A by TOPS_1:19,XBOOLE_1:36;
      Int A c= A by TOPS_1:16;
      then Int A c= A /\ (D \ P) by A4,XBOOLE_1:19;
      then Int A c= A \ {d0} by A2,XBOOLE_1:49;
      hence Int A = A \ {d0} by A5,XBOOLE_0:def 10;
    end;
    hence thesis;
  end;
  assume
A6: the carrier of Y = D;
  assume
A7: for A being Subset of Y st A <> D holds Int A = A \ {d0};
  now
    let A be Subset of Y;
    thus A c= D \ {d0} implies A is open
    proof
      assume
A8:   A c= D \ {d0};
A9:   now
        D /\ {d0} = {d0} by ZFMISC_1:46;
        then
A10:    D meets {d0} by XBOOLE_0:def 7;
        assume A = D;
        then D = D \ {d0} by A8,XBOOLE_0:def 10;
        hence contradiction by A10,XBOOLE_1:83;
      end;
A11:  A = A /\ D by A6,XBOOLE_1:28;
      A = A /\ (D \ {d0}) by A8,XBOOLE_1:28;
      then A = A \ {d0} by A11,XBOOLE_1:49;
      then Int A = A by A7,A9;
      hence thesis;
    end;
    thus A <> D & A is open implies A c= D \ {d0}
    proof
      assume
A12:  A <> D;
      assume A is open;
      then Int A = A by TOPS_1:23;
      then A \ {d0} = A by A7,A12;
      hence thesis by A6,XBOOLE_1:33;
    end;
  end;
  hence thesis by A6,Th25;
end;
