reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;

theorem Th8:
  for X1,X2 being TopSpace st X2 = the TopStruct of X1 holds
  X1 is closed SubSpace of X iff X2 is closed SubSpace of X
proof
  let X1,X2 be TopSpace;
  assume
A1: X2 = the TopStruct of X1;
  thus X1 is closed SubSpace of X implies X2 is closed SubSpace of X
  proof
    assume
A2: X1 is closed SubSpace of X;
    then reconsider Y2 = X2 as SubSpace of X by A1,Th7;
    reconsider A2 = the carrier of Y2 as Subset of X by TSEP_1:1;
    A2 is closed by A1,A2,TSEP_1:11;
    hence thesis by TSEP_1:11;
  end;
  assume
A3: X2 is closed SubSpace of X;
  then reconsider Y1 = X1 as SubSpace of X by A1,Th7;
  reconsider A1 = the carrier of Y1 as Subset of X by TSEP_1:1;
  A1 is closed by A1,A3,TSEP_1:11;
  hence thesis by TSEP_1:11;
end;
