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

theorem Th37:
  for A,B being Element of Open_Domains_of T holds (OPD-Meet T).(A
  ,B) = (D-Meet T).(A,B)
proof
  let A,B be Element of Open_Domains_of T;
A1: A in { D where D is Subset of T : D is open_condensed };
  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
A2: E = B and
A3: E is open_condensed;
  consider D being Subset of T such that
A4: D = A and
A5: D is open_condensed by A1;
  D /\ E is open_condensed by A5,A3,TOPS_1:69;
  then A /\ B is condensed by A4,A2,TOPS_1:67;
  then
A6: A /\ B c= Cl Int(A /\ B) by TOPS_1:def 6;
  thus (OPD-Meet T).(A,B) = A /\ B by Def11
    .= Cl(Int(A0 /\ B0)) /\ (A0 /\ B0) by A6,XBOOLE_1:28
    .= (D-Meet T).(A,B) by Def3;
end;
