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<b>let </b><font color="Maroon">W</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">w</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">W</font>;<br/>



<b>assume </b><a NAME="E1:26"/>
<font color="Maroon">W</font> is <a href="trees_2.html#V1" target="_self">finite-order</a>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:26"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">w</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">W</font>  ex <font color="Olive">B</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="trees_2.html#K1" target="_self">succ</a> <font color="Olive">w</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">B</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:26"/>
 ex <font color="Olive">B</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="trees_2.html#K1" target="_self">succ</a> <font color="Maroon">w</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">B</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:26"/>
 <a href="trees_2.html#K1" target="_self">succ</a> <font color="Maroon">w</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>


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