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 Th18:
  W is finite-order implies succ w is finite
proof
  assume W is finite-order;
  then consider n such that
A1: for w holds ex B being finite set st B = succ w & card B <= n by Th17;
 ex B being finite set st B = succ w & card B <= n by A1;
  hence thesis;
end;
