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

theorem Th32:
  for A,B being Element of Closed_Domains_of T holds (CLD-Union T)
  .(A,B) = (D-Union T).(A,B)
proof
  let A,B be Element of Closed_Domains_of T;
A1: A in { D where D is Subset of T : D is closed_condensed };
  Closed_Domains_of T c= Domains_of T by Th31; then
  reconsider A0 = A, B0 = B as Element of Domains_of T;
  B in { E where E is Subset of T : E is closed_condensed };
  then consider E being Subset of T such that
A2: E = B and
A3: E is closed_condensed;
  consider D being Subset of T such that
A4: D = A and
A5: D is closed_condensed by A1;
  D \/ E is closed_condensed by A5,A3,TOPS_1:68;
  then D \/ E is condensed by TOPS_1:66;
  then
A6: Int Cl(A \/ B) c= A \/ B by A4,A2,TOPS_1:def 6;
  thus (CLD-Union T).(A,B) = A \/ B by Def6
    .= Int(Cl(A0 \/ B0)) \/ (A0 \/ B0) by A6,XBOOLE_1:12
    .= (D-Union T).(A,B) by Def2;
end;
