reserve k, l, m, n, i, j for Nat,
  K, N for non empty Subset of NAT,
  Ke, Ne, Me for Subset of NAT,
  X,Y for set;
reserve f for Function of Segm n,Segm k;
reserve x,y for set;

theorem Th56:
  for N be Subset of NAT st N is finite ex k st for n st n in N holds n<=k
proof
  let N be Subset of NAT;
  assume N is finite;
  then reconsider n = card N as Nat;
A1: N,n are_equipotent by CARD_1:def 2;
  consider F be Function such that
  F is one-to-one and
A2: dom F = n and
A3: rng F = N by A1,WELLORD2:def 4;
  reconsider F as XFinSequence by A2,AFINSQ_1:5;
  reconsider F as XFinSequence of NAT by A3,RELAT_1:def 19;
  reconsider k=Sum F as Element of NAT by ORDINAL1:def 12;
  take k;
  let n such that
A4: n in N;
  F <>0 by A3,A4;
  then
A5: len F>0;
  ex x being object st x in dom F & F.x=n by A3,A4,FUNCT_1:def 3;
  hence thesis by A5,AFINSQ_2:61;
end;
