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 Th32:
  for A being Subset of Y holds A c= MaxADSet(A)
proof
  let A be Subset of Y;
    let x be object;
    assume
A1: x in A;
    then reconsider a = x as Point of Y;
    {a} c= A by A1,ZFMISC_1:31;
    then MaxADSet({a}) c= MaxADSet(A) by Th31;
    then
A2: MaxADSet(a) c= MaxADSet(A) by Th28;
A3: a in {a} by TARSKI:def 1;
    {a} c= MaxADSet(a) by Th12;
    then {a} c= MaxADSet(A) by A2;
    hence x in MaxADSet(A) by A3;
end;
