theorem
  elementary_tree 1 = {{},<*0*>}
proof
 now
    let x be object;
    thus x in {{},<*0*>} implies x in { <*n*> : n < 1 } or x in D
    proof
      assume x in {{},<*0*>};
then    x = {} or x = <*0*> by TARSKI:def 2;
      hence thesis by TARSKI:def 1;
    end;
    assume
A1: x in { <*n*> : n < 1 } or x in D;
 now per cases by A1;
      suppose
     x in { <*n*> : n < 1 };
        then consider n such that
A2:     x = <*n*> and
A3:     n < 1;
     n = 0 by A3,NAT_1:25;
        hence x in {{},<*0*>} by A2,TARSKI:def 2;
      end;
      suppose x in D;
then     x = {} by TARSKI:def 1;
        hence x in {{},<*0*>} by TARSKI:def 2;
      end;
    end;
    hence x in {{},<*0*>};
  end;
  hence thesis by XBOOLE_0:def 3;
end;
