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<b>take </b><font color="Maroon">c<sub>1</sub></font> = 0 <a href="trees_4.html#K9">-tree</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> 0</span>)</span>;<br/>

 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> = 
 <a href="trees_3.html#K14">^</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="trees_4.html#T13">TREES_4:13</a></i>
<br/>.= 
 <a href="trees_1.html#K2">elementary_tree</a> 1
<b>by </b><i><a class="ref" href="trees_3.html#T70">TREES_3:70</a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>
;<br/>
<b>hence </b><a NAME="E2:15_2_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T29">FINSET_1:29</a></i>;<br/>

<b>assume </b><a NAME="E3:15_2_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> 0
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_9.html#D1" target="_self">TREES_9:def 1</a><br/>

<b>then </b> <a href="trees_4.html#K1">root-tree</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> = 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="trees_4.html#T5">TREES_4:5</a></i>
<br/>.= 
0 <a href="trees_4.html#K12">-tree</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> 0</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="trees_4.html#D5">TREES_4:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E5:15_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="trees_4.html#T17">TREES_4:17</a></i>;<br/>


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