theorem Th25:
  for S,X,o,p holds {height t1: t1 in rng p} is natural-membered finite &
  union {height t: t in rng p} is Nat
  proof let S,X,o,p;
    set I = {height t: t in rng p};
    thus
A2: I is natural-membered
    proof
      let a; assume a in I;
      then ex t1 st a = height t1 & t1 in rng p;
      hence thesis;
    end;
    deffunc F(Element of Free(S,X)) = height $1;
A3: rng p is finite;
    thus {F(t1): t1 in rng p} is finite from FRAENKEL:sch 21(A3);
    then reconsider I as finite natural-membered set by A2;
    union I is Nat;
    hence thesis;
  end;
