
theorem Th7:
  for t being finite-branching Tree, p being Element of t for n
  being Nat holds p^<*n*> in succ p iff n < card succ p
proof
  let t be finite-branching Tree, p be Element of t;
  deffunc U(Nat) = p^<*$1*>;
A1: for x being set st x in succ p ex n being Nat st x = U(n)
  proof
    let x be set;
    assume x in succ p;
    then ex n being Nat st x = U(n) & U(n) in t;
    hence thesis;
  end;
A2: for i,j being Nat st i < j & U(j) in succ p holds U(i) in succ p
  proof
    let i,j be Nat;
    assume
A3:    i < j & p^<*j*> in succ p;
     reconsider i,j as Nat;
     p^<*i*> in t by A3,TREES_1:def 3;
    hence thesis;
  end;
A4: for i,j being Nat st U(i) = U(j) holds i = j
  proof
    let i,j be Nat;
    assume p^<*i*> = p^<*j*>;
    hence i = (p^<*j*>).(len p+1) by FINSEQ_1:42
      .= j by FINSEQ_1:42;
  end;
  thus for n being Nat holds U(n) in succ p iff n < card succ p
  from FinOrdSet(A1,A2,A4);
end;
