
theorem
  for T being anti-discrete non empty TopStruct, p being Point of T
  holds {the carrier of T} is Basis of p
proof
  let T be anti-discrete non empty TopStruct, p be Point of T;
  set A = {the carrier of T};
  A c= bool the carrier of T
  proof
    let a be object;
    assume a in A;
    then
A1: a = the carrier of T by TARSKI:def 1;
    the carrier of T c= the carrier of T;
    hence thesis by A1;
  end;
  then reconsider A as Subset-Family of T;
  A is Basis of p
  proof
A2: A is open
    proof
      let a be Subset of T;
      assume a in A;
      then a = [#]T by TARSKI:def 1;
      hence thesis;
    end;
    A is p-quasi_basis
    proof
    Intersect A = meet A by SETFAM_1:def 9
      .= the carrier of T by SETFAM_1:10;
    hence p in Intersect A;
    let S be Subset of T;
    assume S is open & p in S;
    then
A3: S = the carrier of T by TDLAT_3:18;
    take S;
    thus thesis by A3,TARSKI:def 1;
    end;
    hence thesis by A2;
  end;
  hence thesis;
end;
