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

theorem Th10:
  MaxADSF(x) is anti-discrete-set-family
proof
  now
    let A be Subset of Y;
    assume A in MaxADSF(x);
    then ex C being Subset of Y st C = A & C is anti-discrete & x in C;
    hence A is anti-discrete;
  end;
  hence thesis;
end;
