reserve X for non empty set;

theorem Th1:
  for B being 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 FinMeetCl B c= UniCl B
  proof
    let B be empty Subset-Family of X;
    assume that
    for B1,B2 be Element of B
    ex BB being Subset of B st B1/\B2=union BB and
A1: X = union B;
    FinMeetCl B c= UniCl B
    proof
      let x be object;
      assume
A2:   x in FinMeetCl B;
      then reconsider x1=x as Subset of X;
      consider Y be Subset-Family of X such that
A3:   Y c= B and
      Y is finite and
A4:   x1=Intersect Y by A2,CANTOR_1:def 3;
      Y={} & meet {}={} by A3,SETFAM_1:1;
      then x1=X by A4,SETFAM_1:def 9;
      hence x in UniCl B by A1,CANTOR_1:def 1;
    end;
    hence FinMeetCl B c= UniCl B;
  end;
