
theorem
  for T being non empty TopSpace, B being Basis of T, V being Subset of
  T holds Int V = union { G where G is Subset of T: G in B & G c= V }
proof
  let T be non empty TopSpace, B be Basis of T, V be Subset of T;
  set X = { G where G is Subset of T: G in B & G c= V }, Y = { G where G is
  Subset of T: G in B & G c= Int V };
  X = Y
  proof
    thus X c= Y
    proof
      let e be object;
      assume e in X;
      then consider G being Subset of T such that
A1:   e = G and
A2:   G in B and
A3:   G c= V;
      G c= Int V by A2,A3,Th10,TOPS_1:24;
      hence thesis by A1,A2;
    end;
    let e be object;
    assume e in Y;
    then consider G being Subset of T such that
A4: e = G & G in B and
A5: G c= Int V;
    Int V c= V by TOPS_1:16;
    then G c= V by A5;
    hence thesis by A4;
  end;
  hence thesis by Th9;
end;
