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<b>let </b><font color="Maroon">A</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="orders_1.html#M1">Choice_Function</a> of  <a href="orders_1.html#K1">BOOL</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">fC</font> be    <a href="orders_2.html#M2" target="_self">Chain</a> of <font color="Maroon">f</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:58"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC</font>
 <b>and </b><br/><a NAME="E2:58"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:58"/>
( the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="wellord1.html#R2">well_orders</a> <font color="Maroon">fC</font> &amp; <font color="Maroon">fC</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">fC</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">fC</font> )
 
<b>by </b><i><a class="ref" href="orders_2.html#T30" target="_self">Th30</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:58"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC</font>
 <b>and </b><br/><a NAME="E6:58"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T9">WELLORD1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="orders_2.html#K2" target="_self">LowerCone</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">fC</font>;<br/><b>assume </b><a NAME="E2:58_1"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="orders_2.html#K2" target="_self">LowerCone</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">fC</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E3:58_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fC</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="orders_2.html#D11" target="_self">Def11</a>, </i>;<br/><a NAME="E5:58_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E6:58_1"><i><font color="Green">E44</font></i></a>, </i>;<br/><a NAME="E6:58_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><a NAME="E7:58_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K2" target="_self">LowerCone</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="orders_2.html#D11" target="_self">Def11</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:58_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a1</font> where <font color="Olive">a1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : for <font color="Olive">a2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">a2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">a1</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">a2</font> </span>}</span> 
 ;<br/><a NAME="E9:58_1"/><b>then </b><i><font color="Green">E73</font></i>: 
 ex <font color="Olive">a1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a1</font> &amp; ( for <font color="Olive">a2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">a2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">a1</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">a2</font> ) )
 ;<br/><a NAME="E10:58_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E11:58_1"/><b>then </b>
<font color="Maroon">a</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T35" target="_self">Th35</a></i>;<br/><b>hence </b><a NAME="E12:58_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E9:58"/>
<span class="p1">(<span class="default"><a href="orders_2.html#K2" target="_self">LowerCone</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">fC</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#K3" target="_self">InitSegm</a> <font color="Maroon">fC</font>,<font color="Maroon">x</font>
 
;<br/>
<a NAME="E10:58"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="orders_2.html#K1" target="_self">UpperCone</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="orders_2.html#K2" target="_self">LowerCone</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">fC</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E11:58"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:58"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E8:58"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E13:58"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D11" target="_self">Def11</a>, </i>;<br/>


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