now

let x be set ; :: thesis: ( ( not x in {{} ,<*0 *>} or x in { <*n*> where n is Element of NAT : n < 1 } or x in {{} } ) & ( ( x in { <*n*> where n is Element of NAT : n < 1 } or x in {{} } ) implies x in {{} ,<*0 *>} ) )
thus ( not x in {{} ,<*0 *>} or x in { <*n*> where n is Element of NAT : n < 1 } or x in {{} } ) :: thesis: ( ( x in { <*n*> where n is Element of NAT : n < 1 } or x in {{} } ) implies x in {{} ,<*0 *>} )

proof

assume x in {{} ,<*0 *>} ; :: thesis: ( x in { <*n*> where n is Element of NAT : n < 1 } or x in {{} } )
then ( x = {} or x = <*0 *> ) by TARSKI:def 2;
hence ( x in { <*n*> where n is Element of NAT : n < 1 } or x in {{} } ) by TARSKI:def 1; :: thesis: verum

end;

assume A4: ( x in { <*n*> where n is Element of NAT : n < 1 } or x in {{} } ) ; :: thesis: x in {{} ,<*0 *>}

now

per cases ( x in { <*n*> where n is Element of NAT : n < 1 } or x in {{} } ) by A4;

suppose x in { <*n*> where n is Element of NAT : n < 1 } ; :: thesis: x in {{} ,<*0 *>}

then consider n being Element of NAT such that
A7: x = <*n*> and
A8: n < 1 ;
n = 0 by A8, NAT_1:26;
hence x in {{} ,<*0 *>} by A7, TARSKI:def 2; :: thesis: verum

end;

suppose x in {{} } ; :: thesis: x in {{} ,<*0 *>}

then x = {} by TARSKI:def 1;
hence x in {{} ,<*0 *>} by TARSKI:def 2; :: thesis: verum

end;

end;

end;

hence x in {{} ,<*0 *>} ; :: thesis: verum

end;

hence elementary_tree 1 = {{} ,<*0 *>} by XBOOLE_0:def 3; :: thesis: verum