reserve x,y,z,a,b,c,X,X1,X2,Y,Z for set,
  W,W1,W2 for Tree,
  w,w9 for Element of W,
  f for Function,
  D,D9 for non empty set,
  i,k,k1,k2,l,m,n for Nat,
  v,v1,v2 for FinSequence,
  p,q,r,r1,r2 for FinSequence of NAT;

theorem Th17:
  W is finite-order implies ex n st for w holds
  ex B being finite set st B = succ w & card B <= n
proof
  given n such that
A1: for w holds not w^<*n*> in W;
A2: Seg n is finite;
  take n;
  let w;
  deffunc U(Real) = w^<*$1-1*>;
A3: { U(r) where r is Element of REAL: r in Seg n }
    is finite from FRAENKEL:sch 21(A2
  );
A4: succ w c= { w^<*r-1*> where r is Element of REAL: r in Seg n }
  proof
    let x be object;
    assume x in succ w;
    then consider k such that
A5: x = w^<*k*> and
A6: w^<*k*> in W;
 not w^<*n*> in W by A1;
then  k < n by A6,TREES_1:def 3;
then A7: k+1 <= n by NAT_1:13;
 1 <= k+1 by NAT_1:11;
then A8: k+1 in Seg n by A7,FINSEQ_1:1;
A9: k+1 in REAL by XREAL_0:def 1;
 (k+1)-1 = k;
    hence thesis by A5,A8,A9;
  end;
  then reconsider B = succ w as finite set by A3;
  take B;
  thus B = succ w;
  set C = { U(r) where r is Element of REAL: r in Seg n };
 C is finite from FRAENKEL:sch 21(A2);
  then reconsider C as finite set;
A10: C = { U(r) where r is Element of REAL: r in Seg n };
 card C <= card Seg n from FraenkelFinCard(A10);
then A11: card C <= n by FINSEQ_1:57;
 card B <= card C by A4,NAT_1:43;
  hence thesis by A11,XXREAL_0:2;
end;
