reserve n,m,k for Nat,
  X,Y,Z for set,
  f for Function of X,Y,
  H for Subset of X;

theorem Th6:
  for X being finite set st card X < n holds the_subsets_of_card(n, X) is empty
proof
  let X be finite set;
  assume
A1: card X < n;
A2: card Seg n = n by FINSEQ_1:57;
  assume the_subsets_of_card(n, X) is not empty;
  then consider x be object such that
A3: x in the_subsets_of_card(n, X) by XBOOLE_0:def 1;
  ex X9 be Subset of X st x=X9 & card X9 = n by A3;
  then Segm card Seg n c= Segm card X by A2,CARD_1:11;
  then card Seg n <= card X by NAT_1:39;
  hence contradiction by A1,FINSEQ_1:57;
end;
