reserve n for Nat,
        X for set,
        Fx,Gx for Subset-Family of X;

theorem Th3:
  (for G be finite Subset-Family of X st G c=Fx & n+1<card G holds
     meet G is empty) implies order Fx <= n
proof
  set n1=n+1;
  assume
A1: for G be finite Subset-Family of X st G c=Fx & n+1<card G holds
  meet G is empty;
A2: now
A3: card Segm(n1+1)=n1+1;
    let Gx such that
A4: Gx c=Fx and
A5: n1 in card Gx;
    nextcard card Segm n1=card(n1+1) by A3,NAT_1:42;
    then card(n1+1)c=card Gx by A5,CARD_3:90;
    then consider f be Function such that
A6: f is one-to-one & dom f=(n1+1) and
A7: rng f c=Gx by CARD_1:10;
    reconsider R=rng f as Subset-Family of X by A7,XBOOLE_1:1;
    n1+1,R are_equipotent by A6,WELLORD2:def 4;
    then
A8: n1+1=card R by A3,CARD_1:5;
    then reconsider R as finite Subset-Family of X;
    n1<card R by A8,NAT_1:13;
    then
A9: meet R is empty by A1,A4,A7,XBOOLE_1:1;
    R is non empty by A8;
    then meet Gx c={} by A7,A9,SETFAM_1:6;
    hence meet Gx is empty;
  end;
  then reconsider f=Fx as finite-order Subset-Family of X by Def1;
  consider Gx such that
A10: Gx c=f and
A11: card Gx=order f+1 and
A12: meet Gx is non empty or Gx is empty by Def2;
  assume order Fx>n;
  then order f+1>n1 by XREAL_1:6;
  then n1 in Segm(order f + 1) by NAT_1:44;
  hence thesis by A2,A10,A12,A11;
end;
