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
  for T1,T2 being finite Tree holds
  height tree(T1,T2) = max(height T1, height T2)+1
proof
  let T1,T2 be finite Tree;
  set m = max(height T1, height T2);
A1: rng <*T1,T2*> = {T1,T2} by FINSEQ_2:127;
A2: T1 in {T1, T2} by TARSKI:def 2;
A3: T2 in {T1,T2} by TARSKI:def 2;
A4: m = height T1 or m = height T2 by XXREAL_0:def 10;
  now
    let t be finite Tree;
    assume t in rng <*T1,T2*>;
    then t = T1 or t = T2 by A1,TARSKI:def 2;
    hence height t <= m by XXREAL_0:25;
  end;
  hence thesis by A1,A2,A3,A4,Th79;
end;
