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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="orders_1.html#NM2">Order</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:12"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">R</font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</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> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E4:12"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E5:12"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">i</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="E7:12"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span>
 <b>and </b><br/><a NAME="E8:12"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E9:12"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:12"/><b>then </b><i><font color="Green">E69</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E11:12"/>
not  <a href="triang_1.html#K2">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>
<a NAME="E12:12"/><b>then </b><i><font color="Green">E70</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E13:12"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , , , <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>
<a NAME="E14:12"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E15:12"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E16:12"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<a NAME="E17:12"/><b>then </b>
<font color="Maroon">R</font> <a href="relat_2.html#R4">is_antisymmetric_in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a></i>;<br/>
<b>hence </b><a NAME="E18:12"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a></i>;<br/>


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