
theorem
  for X being set, A being Subset-Family of X, a, b being set st a in
  FinMeetCl A & b in FinMeetCl A holds a /\ b in FinMeetCl A
proof
  let X be set, A be Subset-Family of X, a, b be set;
  assume
A1: a in FinMeetCl A & b in FinMeetCl A;
  then reconsider M = {a,b} as Subset-Family of X by ZFMISC_1:32;
  reconsider M as Subset-Family of X;
  a /\ b = meet M by SETFAM_1:11;
  then
A2: a /\ b = Intersect M by SETFAM_1:def 9;
  M c= FinMeetCl A by A1,ZFMISC_1:32;
  then Intersect M in FinMeetCl FinMeetCl A by Def3;
  hence thesis by A2,Th11;
end;
