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

<b>set </b><font color="Maroon">I</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/>
<b>consider </b><font color="Maroon">R1</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E2:3_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">R1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>
<b>set </b><font color="Maroon">f</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">R1</font>;<br/>
<a NAME="E3:3_1_1"/><i><font color="Green">E42</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">R1</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; ( 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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> holds <br/><span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">R1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R1</font> ) )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">R1</font> as   <a href="struct_0.html#NM8">ManySortedSet</a> of <font color="Maroon">R</font> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>set </b><font color="Maroon">f1</font> = <font color="Maroon">f</font>;<br/>
<a NAME="E5:3_1_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">s1</font>, <font color="Olive">s2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s2</font> holds <br/>for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font> holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">s2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<b>assume </b><a NAME="E1:3_1_1_1"/>
<font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s2</font>
 ;<br/>

<i><font color="Green">E45</font></i>: <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> = 
<font color="Maroon">R1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font>
<b>by </b><i/>
;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:3_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> )
 ;<br/>

<b>thus </b><a NAME="E4:3_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:3_1_1"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:3_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:3_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:3_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T25">RELSET_1:25</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f2</font> = <font color="Maroon">f</font> as    <a href="msualg_4.html#M1">ManySortedRelation</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> <b>by </b><i><a class="ref" href="msualg_4.html#D2">MSUALG_4:def 2</a></i>;<br/>
<a NAME="E8:3_1_1"/>
<font color="Maroon">f2</font> is <a href="osalg_4.html#V1" target="_self">os-compatible</a>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:3_1_1"/>
( <font color="Maroon">f</font> is    <a href="msualg_4.html#M1">ManySortedRelation</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">f</font> is <a href="osalg_4.html#V1" target="_self">os-compatible</a> )
 ;<br/>


</div>
