
theorem Th4:
  for X being set for A being Subset-Family of X holds A c= FinMeetCl A
proof
  let X be set;
  let A be Subset-Family of X;
  let x be object;
  assume
A1: x in A;
  then reconsider x9=x as Subset of X;
  reconsider s = {x9} as Subset-Family of X by SUBSET_1:41;
  x = meet s by SETFAM_1:10;
  then
A2: x = Intersect s by SETFAM_1:def 9;
  s c= A by A1,ZFMISC_1:31;
  hence thesis by A2,Def3;
end;
