
theorem Th13:
  for X being set, A being Subset-Family of X, a, b being set st a
  c= FinMeetCl A & b c= FinMeetCl A holds INTERSECTION(a,b) c= FinMeetCl A
proof
  let X be set;
  let A be Subset-Family of X;
  let a, b be set such that
A1: a c= FinMeetCl A & b c= FinMeetCl A;
  let Z be object;
  assume Z in INTERSECTION(a,b);
  then consider V, B being set such that
A2: V in a & B in b and
A3: Z = V /\ B by SETFAM_1:def 5;
  V in FinMeetCl A & B in FinMeetCl A by A1,A2;
  then reconsider M = {V,B} as Subset-Family of X by ZFMISC_1:32;
  reconsider M as Subset-Family of X;
  V /\ B = meet M by SETFAM_1:11;
  then
A4: V /\ B = Intersect M by SETFAM_1:def 9;
  M c= FinMeetCl A by A1,A2,ZFMISC_1:32;
  then Intersect M in FinMeetCl FinMeetCl A by Def3;
  hence thesis by A3,A4,Th11;
end;
