<?xml version="1.0"?>
<div class="add">

<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="E1:24"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">R</font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">A</font>
 ;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="triang_1.html#K2" target="_self">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font>;<br/>
<b>consider </b><font color="Maroon">s1'</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:24"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">s1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">R</font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">s1'</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:24"/>
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="triang_1.html#K2" target="_self">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:24"/><b>then </b><i><font color="Green">E37</font></i>: 
(  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span></span>)</span>,<font color="Maroon">A</font> <a href="tarski.html#R2">are_equipotent</a>  &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span></span>)</span> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E6:24"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span></span>)</span>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T75">FINSEQ_1:75</a></i>;<br/>
<a NAME="E7:24"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>
<a NAME="E8:24"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/>
<b>hence </b><a NAME="E9:24"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> <font color="Maroon">R</font>,<font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>


</div>
