
theorem Th26:
  for T be TopStruct for X be Subset of T st X is open for B be
finite Subset-Family of T st B is Basis of T for Y be set st Y in Components(B)
  holds X misses Y or Y c= X
proof
  let T be TopStruct;
  let X be Subset of T;
  assume X is open;
  then
A1: X in the topology of T by PRE_TOPC:def 2;
  let B be finite Subset-Family of T;
  assume B is Basis of T;
  then the topology of T c= UniCl B by CANTOR_1:def 2;
  then consider Z be Subset-Family of T such that
A2: Z c= B and
A3: X = union Z by A1,CANTOR_1:def 1;
  let Y be set;
  consider p be FinSequence of bool the carrier of T such that
  len p = card B and
A4: rng p = B and
A5: Components(B) = { Intersect (rng MergeSequence(p,q)) where q is
  FinSequence of BOOLEAN : len q = len p } by Def2;
  assume Y in Components(B);
  then consider q be FinSequence of BOOLEAN such that
A6: Y = Intersect (rng MergeSequence(p,q)) and
  len q = len p by A5;
  assume X /\ Y <> {};
  then consider x be object such that
A7: x in X /\ Y by XBOOLE_0:def 1;
  x in X by A7,XBOOLE_0:def 4;
  then consider b be set such that
A8: x in b and
A9: b in Z by A3,TARSKI:def 4;
A10: x in Y by A7,XBOOLE_0:def 4;
A11: Y c= b
  proof
    let z be object;
    consider i be Nat such that
A12: i in dom p and
A13: p.i = b by A4,A2,A9,FINSEQ_2:10;
A14: i in dom MergeSequence(p,q) by A12,Th1;
    now
      per cases by XBOOLEAN:def 3;
      suppose
        q.i = TRUE;
        hence MergeSequence(p,q).i = b by A13,Th2;
      end;
      suppose
        q.i = FALSE;
        then
A15:    MergeSequence(p,q).i = (the carrier of T) \ b by A12,A13,Th3;
        MergeSequence(p,q).i in rng MergeSequence(p,q) by A14,FUNCT_1:def 3;
        then Y c= (the carrier of T) \ b by A6,A15,MSSUBFAM:2;
        hence MergeSequence(p,q).i = b by A10,A8,XBOOLE_0:def 5;
      end;
    end;
    then
A16: b in rng MergeSequence(p,q) by A14,FUNCT_1:def 3;
    assume z in Y;
    hence thesis by A6,A16,SETFAM_1:43;
  end;
  b c= X by A3,A9,ZFMISC_1:74;
  hence thesis by A11;
end;
