theorem Th1:
  branchdeg v = 1 implies v^<*0*> in dom p
proof
  assume branchdeg v = 1;
  then
A1: succ v <> {} by CARD_1:27,TREES_2:def 12;
  set x = the Element of succ v;
  x in succ v by A1;
  then x in {v^<*n*> where n is Nat: v^<*n*> in dom p}
  by TREES_2:def 5;
  then consider n being Nat such that x = v^<*n*> and
A2: v^<*n*> in dom p;
  thus thesis by A2,TREES_1:def 3;
end;
