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<b>let </b><font color="Maroon">T</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">C</font> be    <a href="trees_2.html#M1">Chain</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">t</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:21"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; not <font color="Maroon">C</font> is <a href="finset_1.html#V1">finite</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">t'</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font>  holds <br/>(  not <font color="Olive">t'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> or  not <font color="Maroon">t</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">t'</font> )
 ;<br/><a NAME="E2:21_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">t'</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font>  st <font color="Olive">t'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/><font color="Olive">t'</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t'</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">t'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1_1"/><i><font color="Green">E35</font></i>: 
not <font color="Maroon">t'</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t</font>
 ;<br/><a NAME="E2:21_1_1_1"/>
<font color="Maroon">t</font>,<font color="Maroon">t'</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i>, <a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a>, <a class="ref" href="trees_2.html#D3">TREES_2:def 3</a></i>;<br/><a NAME="E3:21_1_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t'</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/><a NAME="E4:21_1_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">t'</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E5:21_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_1_1"/>
<font color="Maroon">t'</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:21_1"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">t</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">t</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">t'</font> = <font color="Maroon">x</font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:21_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">t'</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:21"><i><font color="Green">E22</font></i></a>, <a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E4:21_1_2"/>
<font color="Maroon">t'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">t</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">t</font></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_2_1_1"/>
( <font color="Maroon">t'</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">t</font> or <font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_2_1_1_1"/>
<font color="Maroon">t'</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">t</font>
 ;<br/><div class="add"><a NAME="E2:21_1_2_1_1_1"/><b>then </b>
<font color="Maroon">t'</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="trees_1.html#T36">TREES_1:36</a></i>;<br/><b>hence </b><a NAME="E3:21_1_2_1_1_1"/>
<font color="Maroon">t'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">t</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">t</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_2_1_1_2"/>
<font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 ;<br/><div class="add"><a NAME="E2:21_1_2_1_1_2"/><b>then </b>
<font color="Maroon">t'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">t</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:21_1_2_1_1_2"/>
<font color="Maroon">t'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">t</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">t</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:21_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">t</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">t</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:21_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21"/>
 ex <font color="Olive">t'</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">t'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">t</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">t'</font> )
 ;<br/>


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