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 Th45:
  for A, B being Element of Domains_of Y holds (D-Union Y).(A,B) =
  A \/ B & (D-Meet Y).(A,B) = A /\ B
proof
  let A, B be Element of Domains_of Y;
  reconsider A0 = A, B0 = B as Element of Closed_Domains_of Y by Th39;
  (D-Union Y).(A,B) = (CLD-Union Y).(A0,B0) by Th40;
  hence (D-Union Y).(A,B) = A \/ B by TDLAT_1:def 6;
  reconsider A0 = A, B0 = B as Element of Open_Domains_of Y by Th42;
  (D-Meet Y).(A,B) = (OPD-Meet Y).(A0,B0) by Th43;
  hence thesis by TDLAT_1:def 11;
end;
