reserve X for non empty set;

theorem
  for B being non empty Subset-Family of X st
  (for B1,B2 being Element of B holds
  ex BB being Subset of B st B1/\B2=union BB) & X = union B
  holds
    ex ET being FMT_TopSpace st the carrier of ET = X & B is Basis of ET
  proof
    let B be non empty Subset-Family of X such that
A1: (for B1,B2 be Element of B
    ex BB being Subset of B st B1/\B2=union BB) and
A2: X = union B;
    set O=UniCl B;
    set T=TopStruct(#X,O#);
    T is TopSpace by A1,A2,Th3;
    then consider ET be FMT_TopSpace such that
A3: the carrier of T=the carrier of ET and
A4: Family_open_set(ET)=the topology of T by Th12;
    reconsider B1=B as Subset-Family of ET by A3;
A5: B1 is open by A4,CANTOR_1:1;
    B1 is quasi_basis by A3,A4;
    hence thesis by A3,A5;
  end;
