
theorem Th9:
  for T being TopStruct, B being Basis of T, V being Subset of T
  st V is open holds V = union { G where G is Subset of T: G in B & G c= V }
proof
  let T be TopStruct, B be Basis of T, V be Subset of T such that
A1: V is open;
  set X = { G where G is Subset of T: G in B & G c= V };
A2: the topology of T c= UniCl B by CANTOR_1:def 2;
  for x being object holds x in V iff ex Y being set st x in Y & Y in X
  proof
    let x be object;
    hereby
      V in the topology of T by A1;
      then consider Y being Subset-Family of T such that
A3:   Y c= B and
A4:   V = union Y by A2,CANTOR_1:def 1;
      assume x in V;
      then consider W being set such that
A5:   x in W and
A6:   W in Y by A4,TARSKI:def 4;
      take W;
      thus x in W by A5;
      reconsider G = W as Subset of T by A6;
      G c= V by A4,A6,ZFMISC_1:74;
      hence W in X by A3,A6;
    end;
    given Y being set such that
A7: x in Y and
A8: Y in X;
    ex G being Subset of T st Y = G & G in B & G c= V by A8;
    hence thesis by A7;
  end;
  hence thesis by TARSKI:def 4;
end;
