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

<b>set </b><font color="Maroon">D</font> =  <a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> 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">a<sub>1</sub></font> &amp;  <a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> );<br/>
<a NAME="E1:39_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <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> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <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>
 ;<br/>

<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</font> as    <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>by </b><i/>;<br/>
<b>take </b>
 <a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">t</font>
;<br/>

<b>thus </b><a NAME="E3:39_1_1_1"/>
 <a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>take </b>
<font color="Maroon">t</font>
;<br/>

<b>thus </b><a NAME="E4:39_1_1_1"/>
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">t</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:39_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <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> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">f</font> <a href="lang1.html#K15">*</a> <font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E4:39_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lang1.html#K15">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="alg_1.html#T1">ALG_1:1</a></i>;<br/>

<a NAME="E5:39_1_1"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <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>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>let </b><font color="Maroon">y</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E6:39_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E7:39_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lang1.html#K15">*</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="alg_1.html#T1">ALG_1:1</a></i>;<br/>
<a NAME="E8:39_1_1"/>
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> as    <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>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">t1</font>
;<br/>

<b>thus </b><a NAME="E10:39_1_1"/>
<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">y</font>
 ;<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="E12:39_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> &amp;  <a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E13:39_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lang1.html#K15">*</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</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>, , <a class="ref" href="alg_1.html#T1">ALG_1:1</a></i>;<br/>


</div>
