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

theorem Th25:
  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
(A c= D \ {d0} implies A is open) & (A <> D & A is open implies A c= D \ {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 holds (A c= D \ {d0} implies A is open) & (A <>
  D & A is open implies A c= D \ {d0})
  by Th22,TOPS_3:76;
  assume
A1: the carrier of Y = D;
  assume
A2: for A being Subset of Y holds (A c= D \ {d0} implies A is open) & (A
  <> D & A is open implies A c= D \ {d0});
  now
    let A be Subset of Y, C be Subset of STS(D,d0);
    assume
A3: A = C;
A4: now
      assume
A5:   A is open;
      now
        per cases;
        case
          A = D;
          then C = [#]STS(D,d0) by A3;
          hence C is open;
        end;
        case
          A <> D;
          then A c= D \ {d0} by A2,A5;
          hence C is open by A3,Th22;
        end;
      end;
      hence C is open;
    end;
    now
      assume
A6:   C is open;
      now
        per cases;
        case
          C = D;
          then A = [#]Y by A1,A3;
          hence A is open;
        end;
        case
          C <> D;
          then A c= D \ {d0} by A3,A6,Th22;
          hence A is open by A2;
        end;
      end;
      hence A is open;
    end;
    hence A is open iff C is open by A4;
  end;
  hence thesis by A1,TOPS_3:72;
end;
