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 Th79:
  for t being finite Tree st t in rng w &
  for t9 being finite Tree st t9 in rng w holds height t9 <= height t holds
  height tree(w) = (height t) + 1
proof
  let t be finite Tree such that
A1: t in rng w and
A2: for t9 being finite Tree st t9 in rng w holds height t9 <= height t;
A3: height tree(w) > height t by A1,Th78;
A4: height tree(w) <= (height t) + 1 by A2,Th77;
  height tree(w) >= (height t) + 1 by A3,NAT_1:13;
  hence thesis by A4,XXREAL_0:1;
end;
