
theorem
  for T being anti-discrete non empty TopStruct, p being Point of T
  holds {the carrier of T} is correct 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;
  reconsider A as Subset-Family of T;
  A is basis of p
  proof
    let S be a_neighborhood of p;
    take S;
    p in Int S by CONNSP_2:def 1;
    then
A2: Int S = the carrier of T by TDLAT_3:18;
    Int S c= S by TOPS_1:16;
    then Int S = S by A2;
    hence thesis by A2,TARSKI:def 1;
  end;
  then reconsider A as basis of p;
  A is correct
  proof
    let X be Subset of T;
    hereby
      assume X in A;
      then X = the carrier of T by TARSKI:def 1;
      then Int X = [#]T by TOPS_1:15;
      hence p in Int X;
    end;
    assume p in Int X;
    then
A3: Int X = the carrier of T by TDLAT_3:18;
    Int X c= X by TOPS_1:16;
    then Int X = X by A3;
    hence thesis by A3,TARSKI:def 1;
  end;
  hence thesis;
end;
