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

theorem Th33:
  for A,B being Element of Closed_Domains_of T holds (CLD-Meet T).
  (A,B) = (D-Meet T).(A,B)
proof
  let A,B be Element of Closed_Domains_of T;
  A in { D where D is Subset of T : D is closed_condensed };
  then consider D being Subset of T such that
A1: D = A and
A2: D is closed_condensed;
A3: Int(A /\ B) c= A /\ B by TOPS_1:16;
  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
A4: E = B and
A5: E is closed_condensed;
A6: E is closed by A5,TOPS_1:66;
  D is closed by A2,TOPS_1:66;
  then
A7: Cl(D /\ E) = D /\ E by A6,PRE_TOPC:22;
  thus (CLD-Meet T).(A,B) = Cl(Int(A /\ B)) by Def7
    .= Cl(Int(A0 /\ B0)) /\ (A0 /\ B0) by A1,A4,A7,A3,PRE_TOPC:19,XBOOLE_1:28
    .= (D-Meet T).(A,B) by Def3;
end;
