reserve T for 1-sorted;
reserve T for TopSpace;

theorem Th39:
  CLD-Union T = (D-Union T)||Closed_Domains_of T
proof
A1:  Closed_Domains_of T c= Domains_of T by Th31; then
  reconsider F = CLD-Union T as Function of [:Closed_Domains_of T,
  Closed_Domains_of T:],Domains_of T by FUNCT_2:7;
  [:Closed_Domains_of T,Closed_Domains_of T:] c=
    [:Domains_of T,Domains_of T:] by A1,ZFMISC_1:96; then
  reconsider G = (D-Union T)||Closed_Domains_of T as Function of [:
  Closed_Domains_of T,Closed_Domains_of T:],Domains_of T by FUNCT_2:32;
  for A being Element of Closed_Domains_of T, B being Element of
  Closed_Domains_of T holds F.(A,B) = G.(A,B)
  proof
    let A be Element of Closed_Domains_of T, B be Element of Closed_Domains_of
    T;
    thus F.(A,B) = (D-Union T).(A,B) by Th32
      .= ((D-Union T)||Closed_Domains_of T). [A,B] by FUNCT_1:49
      .= G.(A,B);
  end;
  hence thesis by BINOP_1:2;
end;
