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

theorem Th10:
  for X being finite set st k <= card X ex Y being Subset of X st card Y = k
proof
  let X be finite set;
  assume k <= card X;
  then Segm card k c= Segm card X by NAT_1:39;
  then consider Y be set such that
A1: Y c= X and
A2: card Y = card k by CARD_FIL:36;
  reconsider Y as Subset of X by A1;
  take Y;
  thus thesis by A2;
end;
