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

theorem Th36:
  for A,B being Element of Open_Domains_of T holds (OPD-Union T).(
  A,B) = (D-Union T).(A,B)
proof
  let A,B be Element of Open_Domains_of T;
  A in { D where D is Subset of T : D is open_condensed };
  then consider D being Subset of T such that
A1: D = A and
A2: D is open_condensed;
A3: A \/ B c= Cl(A \/ B) by PRE_TOPC:18;
  Open_Domains_of T c= Domains_of T by Th35; then
  reconsider A0 = A, B0 = B as Element of Domains_of T;
  B in { E where E is Subset of T : E is open_condensed };
  then consider E being Subset of T such that
A4: E = B and
A5: E is open_condensed;
A6: E is open by A5,TOPS_1:67;
  D is open by A2,TOPS_1:67;
  then
A7: Int(D \/ E) = D \/ E by A6,TOPS_1:23;
  thus (OPD-Union T).(A,B) = Int(Cl(A \/ B)) by Def10
    .= Int(Cl(A0 \/ B0)) \/ (A0 \/ B0) by A1,A4,A7,A3,TOPS_1:19,XBOOLE_1:12
    .= (D-Union T).(A,B) by Def2;
end;
