reserve
  X,x,y,z for set,
  k,n,m for Nat ,
  f for Function,
  p,q,r for FinSequence of NAT;
reserve p1,p2,p3 for FinSequence;
reserve T,T1 for Tree;
reserve fT,fT1 for finite Tree;
reserve t for Element of T;
reserve w for FinSequence;
reserve t1,t2 for Element of T;
reserve s,t for FinSequence;

theorem
  elementary_tree 2 = {{},<*0*>,<*1*>}
proof
 now
    let x be object;
    thus
    x in {{},<*0*>,<*1*>} implies x in { <*n*> : n < 2 } or x in D
    proof
      assume x in {{},<*0*>,<*1*>};
then    x = {} or x = <*0*> or x = <*1*> by ENUMSET1:def 1;
      hence thesis by TARSKI:def 1;
    end;
    assume
A1: x in { <*n*> : n < 2 } or x in D;
 now per cases by A1;
      suppose
     x in { <*n*> : n < 2 };
        then consider n such that
A2:     x = <*n*> and
A3:     n < 2;
     n = 0 or ... or n = 2 by A3;
        hence x in {{},<*0*>,<*1*>} by A2,A3,ENUMSET1:def 1;
      end;
      suppose x in D;
then     x = {} by TARSKI:def 1;
        hence x in {{},<*0*>,<*1*>} by ENUMSET1:def 1;
      end;
    end;
    hence x in {{},<*0*>,<*1*>};
  end;
  hence thesis by XBOOLE_0:def 3;
end;
