reserve T for TopSpace;
reserve T for non empty TopSpace;
reserve F for Subset-Family of T;
reserve T for non empty TopSpace;

theorem Th64:
  for F being Subset-Family of T holds F c= Domains_of T iff F is
  domains-family
proof
  let F be Subset-Family of T;
  thus F c= Domains_of T implies F is domains-family
  proof
    assume
A1: F c= Domains_of T;
    now
      let A be Subset of T;
      assume A in F;
      then A in Domains_of T by A1;
      then A in {P where P is Subset of T : P is condensed} by TDLAT_1:def 1;
      then ex Q being Subset of T st Q = A & Q is condensed;
      hence A is condensed;
    end;
    hence thesis;
  end;
  thus F is domains-family implies F c= Domains_of T
  proof
    assume
A2: F is domains-family;
    for X being object holds X in F implies X in Domains_of T
    proof
      let X be object;
      assume
A3:   X in F;
      then reconsider X0 = X as Subset of T;
      X0 is condensed by A2,A3;
      then X0 in {P where P is Subset of T : P is condensed};
      hence thesis by TDLAT_1:def 1;
    end;
    hence thesis;
  end;
end;
