reserve T, T1 for Tree,
  P for AntiChain_of_Prefixes of T,
  p1 for FinSequence,
  p, q, r, s, p9 for FinSequence of NAT,
  x, Z for set,
  t for Element of T,
  k, n for Nat;

theorem Th9:
  tree(T,{t},T1) = T with-replacement (t,T1)
proof
  let p;
  thus p in tree(T,{t},T1) implies p in T with-replacement (t,T1)
  proof
    assume
A1: p in tree(T,{t},T1);
A2: now
      assume
A3:   p in T & for s st s in {t} holds not s is_a_proper_prefix_of p;
   t in {t} by TARSKI:def 1;
      hence p in T & not t is_a_proper_prefix_of p by A3;
    end;
 now
      given s such that
A4:   ex r st s in {t} & r in T1 & p = s^r;
   s = t by A4,TARSKI:def 1;
      hence ex r st r in T1 & p = t^r by A4;
    end;
    hence thesis by A1,A2,Def1,TREES_1:def 9;
  end;
  assume
A5: p in T with-replacement (t,T1);
A6: p in T & not t is_a_proper_prefix_of p implies
  p in T & for s st s in {t} holds not s is_a_proper_prefix_of p
  by TARSKI:def 1;
 now
    assume
A7: ex r st r in T1 & p = t^r;
    thus ex s,r st s in {t} & r in T1 & p = s^r
    proof
      take t;
  t in {t} by TARSKI:def 1;
      hence thesis by A7;
    end;
  end;
  hence thesis by A5,A6,Def1,TREES_1:def 9;
end;
