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<b>take </b><font color="Maroon" title="c1">t</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="trees_4.html#K9" title="TREES_4:func.9">-tree</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">t</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; not <font color="Maroon" title="c1">t</font> is <a href="trees_9.html#V1" title="TREES_9:attr.1">root</a> )</span><br/>

 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">t</font> = 
 <a href="trees_3.html#K14" title="TREES_3:func.14">^</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="trees_4.html#T13" title="TREES_4:th.13">TREES_4:13</a></i>
<br/>.= 
 <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 1
<b>by </b><i><a class="ref" href="trees_3.html#T70" title="TREES_3:th.70">TREES_3:70</a>, <a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></i>
;<br/>
<b>hence </b><a NAME="E2:15_2_1"/>
<font color="Maroon" title="c1">t</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T29" title="FINSET_1:th.29">FINSET_1:29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c1">t</font> is <a href="trees_9.html#V1" title="TREES_9:attr.1">root</a></span><br/>

<b>assume </b><a NAME="E3:15_2_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_9.html#D1" target="_self" title="TREES_9:def.1">TREES_9:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then </b> <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> = 
<font color="Maroon" title="c1">t</font>
<b>by </b><i><a class="ref" href="trees_4.html#T5" title="TREES_4:th.5">TREES_4:5</a></i>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="trees_4.html#D5" title="TREES_4:def.5">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" title="TREES_4:th.17">TREES_4:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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