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

theorem
  order Fx <= n implies
    for Gx st Gx c= Fx & n+1 in card Gx holds meet Gx is empty
proof
A1: card(n+1)=n+1;
  assume
A2: order Fx<=n;
  then reconsider f=Fx as finite-order Subset-Family of X by Th1;
  order f+1<=n+1 by A2,XREAL_1:6;
  then order f+1<n+1+1 by NAT_1:13;
  then
A3: order f+1 in Segm(n+2) by NAT_1:44;
  let Gx such that
A4: Gx c=Fx and
A5: n+1 in card Gx;
  nextcard Segm(n+1)c=card Gx by A5,CARD_3:90;
  then card Segm(n+1+1)c=card Gx by A1,NAT_1:42;
  hence thesis by A4,A3,Def2;
end;
