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

theorem Th79:
  for F being Subset-Family of T holds F is open-domains-family
  implies F is domains-family
proof
  let F be Subset-Family of T;
  thus F is open-domains-family implies F is domains-family
  proof
    assume F is open-domains-family;
    then
A1: F c= Open_Domains_of T by Th78;
    Open_Domains_of T c= Domains_of T by TDLAT_1:35;
    then F c= Domains_of T by A1;
    hence thesis by Th64;
  end;
end;
