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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="E1:23"/><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="E3:23"/><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="E4:23"/>
 <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>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </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> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM7">FinSequence</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">A</font></span>)</span> #) ;<br/>
<a NAME="E7:23"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">l</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font>, <font color="Maroon">l</font> = <font color="Maroon">l</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">fk</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">A</font></span>)</span> #) <b>by </b><i>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fl</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">A</font></span>)</span> #) <b>by </b><i>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E5:23_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">fl</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">l</font>
 ;<br/><i><font color="Green">E41</font></i>: <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<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="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><i><font color="Green">E42</font></i>: <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<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="finseq_4.html#K4">/.</a> <font color="Maroon">l</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_3"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font> or <font color="Maroon">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1"><b>suppose </b></a><a NAME="E1:23_1_1_3_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_1_3_1"/>
contradiction
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_2"><b>suppose </b></a><a NAME="E1:23_1_1_3_2"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_1_3_2"/>
contradiction
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:23_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:23"/>
 <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>
 ;<br/>


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