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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be  <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>assume </b><a NAME="E1:19"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">R</font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">A</font>
 ;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:19"/><i><font color="Green">E34</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> &amp; <font color="Maroon">j</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="E3:19"/><i><font color="Green">E35</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> <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">j</font>
 ;<br/>

<b>assume </b><a NAME="E4:19"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E5:19"/><b>then </b>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> or <font color="Maroon">j</font> <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/>
<b>hence </b><a NAME="E6:19"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>


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