<?xml version="1.0"?>
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<b>set </b><font color="Maroon">D</font> =  <a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font>;<br/>
<b>let </b><font color="Maroon">f1</font> be    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a> , <font color="Maroon">f2</font> be    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:39_1_2"/><i><font color="Green">E62</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> holds <br/> ex <font color="Olive">t</font> being   <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> st <br/>( <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Olive">t</font> ) ) )
 <b>and </b><br/><a NAME="E2:39_1_2"/><i><font color="Green">E64</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> holds <br/> ex <font color="Olive">t</font> being   <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> st <br/>( <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Olive">t</font> ) ) )
 ;<br/>

<a NAME="E3:39_1_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font> holds <br/><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_2_1"/><i><font color="Green">E65</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">f1</font>
 ;<br/>

<b>consider </b><font color="Maroon">t1</font> being    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E3:39_1_2_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">t1</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">t2</font> being    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E5:39_1_2_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">t2</font> )
 <b>by </b><i>, , </i>;<br/>
<b>thus </b><a NAME="E6:39_1_2_1"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:39_1_2"/>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


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