reserve X for non empty set;

theorem
  for ET being FMT_TopSpace, B being Basis of ET holds
  (for B1,B2 being Element of B holds
  ex BB being Subset of B st B1 /\ B2 = union BB) &
  (the carrier of ET = union B)
  proof
    let X be FMT_TopSpace, B be Basis of X;
A1: B is quasi_basis;
    thus (for B1,B2 be Element of B holds
    ex BB being Subset of B st B1/\B2=union BB)
    proof
      let B1,B2 be Element of B;
      per cases;
      suppose B is empty; then
A2:     UniCl B ={{}} by YELLOW_9:16;
        the carrier of X in Family_open_set(X) by Th9;
        hence thesis by A1,A2,TARSKI:def 1;
      end;
      suppose
A3:     not B is empty;
        B is open;
        then B c= Family_open_set(X);
        then B1 in Family_open_set(X) & B2 in Family_open_set(X) by A3;
        then B1/\B2 in Family_open_set(X) & B is quasi_basis by Th9;
        then consider Y be Subset-Family of X such that
A4:     Y c= B & B1/\B2 = union Y by CANTOR_1:def 1;
        reconsider Y as Subset of B by A4;
        thus ex BB being Subset of B st B1/\B2=union BB by A4;
      end;
    end;
    the carrier of X in Family_open_set(X) by Th9;
    then consider Y be Subset-Family of X such that
A5: Y c= B & the carrier of X=union Y by A1,CANTOR_1:def 1;
    thus the carrier of X c= union B by A5,ZFMISC_1:77;
    thus union B c= the carrier of X;
  end;
