reserve X for non empty TopSpace;

theorem Th6:
  for M being Subset of X st M is maximal_T_0 for A being Subset of
  X st A is open holds A = MaxADSet(M /\ A)
proof
  let M be Subset of X;
  assume
A1: M is maximal_T_0;
  let A be Subset of X;
  assume
A2: A is open;
  then MaxADSet(M /\ A) = MaxADSet(M) /\ MaxADSet(A) by TEX_4:65;
  then
A3: MaxADSet(M /\ A) = MaxADSet(M) /\ A by A2,TEX_4:56;
  A = (the carrier of X) /\ A by XBOOLE_1:28;
  hence thesis by A1,A3;
end;
