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 Th30:
  for A being Subset of Y, x being Point of Y holds MaxADSet(x)
  meets MaxADSet(A) implies MaxADSet(x) c= MaxADSet(A)
proof
  let A be Subset of Y, x be Point of Y;
  set F = {MaxADSet(b) where b is Point of Y : b in A};
  assume MaxADSet(x) meets MaxADSet(A);
  then MaxADSet(x) meets A by Th29;
  then consider z being object such that
A1: z in MaxADSet(x) /\ A by XBOOLE_0:4;
  reconsider z as Point of Y by A1;
  set E = {MaxADSet(a) where a is Point of Y : a in {z}};
  z in A by A1,XBOOLE_0:def 4;
  then
A2: {z} c= A by ZFMISC_1:31;
  E c= F
  proof
    let C be object;
    assume C in E;
    then ex a being Point of Y st C = MaxADSet(a) & a in {z};
    hence thesis by A2;
  end;
  then MaxADSet({z}) c= MaxADSet(A) by ZFMISC_1:77;
  then
A3: MaxADSet(z) c= MaxADSet(A) by Th28;
  z in MaxADSet(x) by A1,XBOOLE_0:def 4;
  hence thesis by A3,Th21;
end;
