now :: thesis: for x being object holds
( ( not x in {{},<*0*>,<*1*>} or x in { <*n*> where n is Nat : n < 2 } or x in {{}} ) & ( ( x in { <*n*> where n is Nat : n < 2 } or x in {{}} ) implies x in {{},<*0*>,<*1*>} ) )

let x be object ; :: thesis: ( ( not x in {{},<*0*>,<*1*>} or x in { <*n*> where n is Nat : n < 2 } or x in {{}} ) & ( ( x in { <*n*> where n is Nat : n < 2 } or x in {{}} ) implies x in {{},<*0*>,<*1*>} ) )
thus ( not x in {{},<*0*>,<*1*>} or x in { <*n*> where n is Nat : n < 2 } or x in {{}} ) :: thesis: ( ( x in { <*n*> where n is Nat : n < 2 } or x in {{}} ) implies x in {{},<*0*>,<*1*>} )

proof

assume x in {{},<*0*>,<*1*>} ; :: thesis: ( x in { <*n*> where n is Nat : n < 2 } or x in {{}} )
then ( x = {} or x = <*0*> or x = <*1*> ) by ENUMSET1:def 1;
hence ( x in { <*n*> where n is Nat : n < 2 } or x in {{}} ) by TARSKI:def 1; :: thesis: verum

end;

assume A1: ( x in { <*n*> where n is Nat : n < 2 } or x in {{}} ) ; :: thesis: x in {{},<*0*>,<*1*>}

now :: thesis: x in {{},<*0*>,<*1*>}

per cases ( x in { <*n*> where n is Nat : n < 2 } or x in {{}} ) by A1;

suppose x in { <*n*> where n is Nat : n < 2 } ; :: thesis: x in {{},<*0*>,<*1*>}

then consider n being Nat such that
A2: x = <*n*> and
A3: n < 2 ;
not not n = 0 & ... & not n = 2 by A3;
hence x in {{},<*0*>,<*1*>} by A2, A3, ENUMSET1:def 1; :: thesis: verum

end;

suppose x in {{}} ; :: thesis: x in {{},<*0*>,<*1*>}

then x = {} by TARSKI:def 1;
hence x in {{},<*0*>,<*1*>} by ENUMSET1:def 1; :: thesis: verum

end;

end;

end;

hence x in {{},<*0*>,<*1*>} ; :: thesis: verum

end;

hence elementary_tree 2 = {{},<*0*>,<*1*>} by XBOOLE_0:def 3; :: thesis: verum