reserve X for TopSpace;
reserve C for Subset of X;
reserve A, B for Subset of X;
reserve X for non empty TopSpace;

theorem
  (for X0 being closed non empty SubSpace of X holds X0 is
  extremally_disconnected) implies X is hereditarily_extremally_disconnected
proof
  assume
A1: for Y being closed non empty SubSpace of X holds Y is
  extremally_disconnected;
  for X0 being non empty SubSpace of X holds X0 is extremally_disconnected
  proof
    let X0 be non empty SubSpace of X;
    reconsider A0 = the carrier of X0 as Subset of X by TSEP_1:1;
    set A = Cl A0;
    A is non empty by PCOMPS_1:2;
    then consider Y being strict closed non empty SubSpace of X such that
A2: A = the carrier of Y by TSEP_1:15;
    A0 c= A by PRE_TOPC:18;
    then reconsider Y0 = X0 as non empty SubSpace of Y by A2,TSEP_1:4;
    reconsider B0 = the carrier of Y0 as Subset of Y by TSEP_1:1;
    Cl B0 = A /\ [#]Y by PRE_TOPC:17;
    then
A3: B0 is dense by A2,TOPS_1:def 3;
    Y is extremally_disconnected by A1;
    hence thesis by A3,Th37;
  end;
  hence thesis;
end;
