reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T;
reserve x for Point of T;

theorem
  BorelSets T = sigma Topology_of T
proof
  reconsider BT = BorelSets T as SigmaField of T by Th13;
  set X = Topology_of T;
A1: for Z being set st X c= Z & Z is SigmaField of T holds BT c= Z
  proof
    let Z be set;
    assume that
A2: X c= Z and
A3: Z is SigmaField of T;
    reconsider Z9 = Z as SigmaField of T by A3;
    Z9 is all-open-containing by A2,Th59;
    hence thesis by Def11;
  end;
  X c= BT by Th65;
  hence thesis by A1,PROB_1:def 9;
end;
