reserve x, X, Y for set;
reserve L for complete LATTICE,
  a for Element of L;

theorem
  for L being RelStr for F being Subset-Family of L st for X being
  Subset of L st X in F holds X is upper holds meet F is upper Subset of L
proof
  let L be RelStr;
  let F be Subset-Family of L;
  reconsider F9 = F as Subset-Family of L;
  assume
A1: for X being Subset of L st X in F holds X is upper;
  reconsider M = meet F9 as Subset of L;
  per cases;
  suppose
    F = {};
    then for x, y being Element of L st x in M & x <= y holds y in M by
SETFAM_1:def 1;
    hence thesis by WAYBEL_0:def 20;
  end;
  suppose
A2: F <> {};
    for x, y being Element of L st x in M & x <= y holds y in M
    proof
      let x, y be Element of L;
      assume that
A3:   x in M and
A4:   x <= y;
      for Y being set st Y in F holds y in Y
      proof
        let Y be set;
        assume
A5:     Y in F;
        then reconsider X = Y as Subset of L;
        X is upper & x in X by A1,A3,A5,SETFAM_1:def 1;
        hence thesis by A4;
      end;
      hence thesis by A2,SETFAM_1:def 1;
    end;
    hence thesis by WAYBEL_0:def 20;
  end;
end;
