
theorem
  for T being non empty TopSpace, P being basis of T holds the topology
  of T c= UniCl Int P
proof
  let T be non empty TopSpace, P be basis of T;
  let x be object;
  assume
A1: x in the topology of T;
  then reconsider X = x as Subset of T;
  ex Y being Subset-Family of T st Y c= Int P & X = union Y
  proof
    set Y = {A where A is Subset of T: A in Int P & Int A c= X};
    Y c= bool the carrier of T
    proof
      let y be object;
      assume y in Y;
      then ex A being Subset of T st y = A & A in Int P & Int A c= X;
      hence thesis;
    end;
    then reconsider Y as Subset-Family of T;
    reconsider Y as Subset-Family of T;
    take Y;
    hereby
      let y be object;
      assume y in Y;
      then ex A being Subset of T st y = A & A in Int P & Int A c= X;
      hence y in Int P;
    end;
    hereby
      let y be object;
      assume
A2:   y in X;
      then reconsider p = y as Point of T;
      reconsider C = P as basis of p by Def4;
      X is open by A1;
      then p in Int X by A2,TOPS_1:23;
      then consider W being Subset of T such that
A3:   W in C and
A4:   p in Int W and
A5:   W c= X by Def1;
      Int W c= W by TOPS_1:16;
      then
A6:   Int Int W c= X by A5;
      Int W in Int P by A3,TDLAT_2:def 1;
      then Int W in Y by A6;
      hence y in union Y by A4,TARSKI:def 4;
    end;
    let y be object;
    assume y in union Y;
    then consider K being set such that
A7: y in K and
A8: K in Y by TARSKI:def 4;
    consider A being Subset of T such that
A9: A = K and
A10: A in Int P and
A11: Int A c= X by A8;
    reconsider A as Subset of T;
    ex E being Subset of T st A = Int E & E in P by A10,TDLAT_2:def 1;
    hence thesis by A7,A9,A11;
  end;
  hence thesis by CANTOR_1:def 1;
end;
