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

theorem Th39:
  Domains_of Y = Closed_Domains_of Y
proof
  now
    let S be object;
    assume
A1: S in Domains_of Y;
    then reconsider A = S as Subset of Y;
    A in {D where D is Subset of Y : D is condensed} by A1;
    then ex D being Subset of Y st D = A & D is condensed;
    then A is closed_condensed by Th33;
    then A in {E where E is Subset of Y : E is closed_condensed};
    hence S in Closed_Domains_of Y;
  end;
  then
A2: Domains_of Y c= Closed_Domains_of Y;
  Closed_Domains_of Y c= Domains_of Y by TDLAT_1:31;
  hence thesis by A2;
end;
