
theorem :: LeavesDef3:
  for T being Tree, t being Element of T holds succ t = {} iff t in Leaves T
proof
  let T be Tree, t be Element of T;
  hereby
    assume succ t = {};
    then not t^<*0*> in {t^<*n*> where n is Nat:t^<*n*> in T } by
TREES_2:def 5;
    then not t^<*0*> in T;
    hence t in Leaves T by TREES_1:54;
  end;
  set x = the Element of succ t;
  assume t in Leaves T;
  then
A1: not t^<*0*> in T by TREES_1:54;
  assume succ t <> {};
  then x in succ t;
  then x in { t^<*n*> where n is Nat : t^<*n*> in T} by
TREES_2:def 5;
  then consider n being Nat such that
  x = t^<*n*> and
A2: t^<*n*> in T;
  0 <= n by NAT_1:2;
  hence contradiction by A1,A2,TREES_1:def 3;
end;
