reserve X for non empty TopSpace;
reserve Y for non empty TopStruct;
reserve x for Point of Y;
reserve Y for non empty TopStruct;

theorem Th29:
  for A being Subset of Y, x being Point of Y holds MaxADSet(x)
  meets MaxADSet(A) implies MaxADSet(x) meets A
proof
  let A be Subset of Y, x be Point of Y;
  set E = {MaxADSet(a) where a is Point of Y : a in A};
  assume MaxADSet(x) /\ MaxADSet(A) <> {};
  then consider z being object such that
A1: z in MaxADSet(x) /\ MaxADSet(A) by XBOOLE_0:def 1;
  reconsider z as Point of Y by A1;
  z in MaxADSet(A) by A1,XBOOLE_0:def 4;
  then consider C being set such that
A2: z in C and
A3: C in E by TARSKI:def 4;
  z in MaxADSet(x) by A1,XBOOLE_0:def 4;
  then
A4: MaxADSet(z) = MaxADSet(x) by Th21;
  consider b being Point of Y such that
A5: C = MaxADSet(b) and
A6: b in A by A3;
  MaxADSet(b) = MaxADSet(z) by A2,A5,Th21;
  then {b} c= MaxADSet(x) by A4,Th12;
  then b in MaxADSet(x) by ZFMISC_1:31;
  hence thesis by A6,XBOOLE_0:3;
end;
