reserve x,y,z for object, X,Y for set,
  i,k,n for Nat,
  p,q,r,s for FinSequence,
  w for FinSequence of NAT,
  f for Function;
reserve w for FinTree-yielding FinSequence;

theorem Th78:
  for t being finite Tree st t in rng w holds height tree(w) > height t
proof
  let t be finite Tree;
  assume t in rng w;
  then consider x being object such that
A1: x in dom w and
A2: t = w.x by FUNCT_1:def 3;
  reconsider x as Element of NAT by A1;
A3: 1 <= x by A1,FINSEQ_3:25;
A4: len w >= x by A1,FINSEQ_3:25;
  consider n be Nat such that
A5: x = 1+n by A3,NAT_1:10;
  reconsider n as Element of NAT by ORDINAL1:def 12;
A6: n < len w by A4,A5,NAT_1:13;
A7: {} in t by TREES_1:22;
A8: <*n*>^{} = <*n*> by FINSEQ_1:34;
A9: t = (tree(w))|<*n*> by A2,A5,A6,Th49;
  <*n*> in tree(w) by A2,A5,A6,A7,A8,Def15;
  hence thesis by A9,TREES_1:48;
end;
