theorem Th10:
  P<>{} implies for q st q in dom tree(T,P,T1) holds
  (for p st p in P holds not p is_a_prefix_of q & tree(T,P,T1).q = T.q)
  or ex p,r st p in P & r in dom T1 & q = p^r & tree(T,P,T1).q = T1.r
proof
  assume
A1: P<>{};
  let q;
  assume q in dom tree(T,P,T1);
then  q in tree(dom T,P,dom T1) by A1,Def2;
  hence thesis by Def2;
end;
