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

theorem Th42:
  OPD-Union T = (D-Union T)||Open_Domains_of T
proof
A1: Open_Domains_of T c= Domains_of T by Th35; then
  reconsider F = OPD-Union T as Function of [:Open_Domains_of T,
  Open_Domains_of T:],Domains_of T by FUNCT_2:7;
  [:Open_Domains_of T,Open_Domains_of T:] c=
    [:Domains_of T,Domains_of T:] by A1,ZFMISC_1:96; then
  reconsider G = (D-Union T)||Open_Domains_of T as Function of [:
  Open_Domains_of T,Open_Domains_of T:],Domains_of T by FUNCT_2:32;
  for A being Element of Open_Domains_of T, B being Element of
  Open_Domains_of T holds F.(A,B) = G.(A,B)
  proof
    let A be Element of Open_Domains_of T, B be Element of Open_Domains_of T;
    thus F.(A,B) = (D-Union T).(A,B) by Th36
      .= ((D-Union T)||Open_Domains_of T) . [A,B] by FUNCT_1:49
      .= G.(A,B);
  end;
  hence thesis by BINOP_1:2;
end;
