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 Th33:
  for A being Subset of Y holds MaxADSet(A) = MaxADSet(MaxADSet(A) )
proof
  let A be Subset of Y;
A1: MaxADSet(MaxADSet(A)) c= MaxADSet(A)
  proof
    let x be object;
    assume x in MaxADSet(MaxADSet(A));
    then consider C being set such that
A2: x in C and
A3: C in {MaxADSet(a) where a is Point of Y : a in MaxADSet(A)} by TARSKI:def 4
;
    consider a being Point of Y such that
A4: C = MaxADSet(a) and
A5: a in MaxADSet(A) by A3;
    consider D being set such that
A6: a in D and
A7: D in {MaxADSet(b) where b is Point of Y : b in A} by A5,TARSKI:def 4;
    consider b being Point of Y such that
A8: D = MaxADSet(b) and
    b in A by A7;
    MaxADSet(a) = MaxADSet(b) by A6,A8,Th21;
    hence thesis by A2,A4,A7,A8,TARSKI:def 4;
  end;
  MaxADSet(A) c= MaxADSet(MaxADSet(A)) by Th32;
  hence thesis by A1;
end;
