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

<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">M</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>



<b>set </b><font color="Maroon">J</font> = <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="pboole.html#K10">|]</a></span>;<br/>
<a NAME="E1:3"/>
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> <font color="Maroon">I</font> holds <br/><span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">M</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"><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"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:3_1"/><b>then </b>
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>;<br/>
<b>hence </b><a NAME="E3:3_1"/>
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">J</font> = <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="pboole.html#K10">|]</a></span> as   <a href="msualg_4.html#NM3">ManySortedRelation</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="msualg_4.html#D2">MSUALG_4:def 2</a></i>;<br/>
<a NAME="E3:3"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">R</font> being  <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">R</font> holds <br/><font color="Olive">R</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_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>let </b><font color="Maroon">R</font> be   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>;<br/>



<b>assume </b><a NAME="E1:3_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> )
 ;<br/>

<b>then </b><font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>
<br/>.= 
 <a href="eqrel_1.html#K1">nabla</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="eqrel_1.html#D1">EQREL_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E3:3_2"/>
<font color="Maroon">R</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T7">EQREL_1:7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:3"/>
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="pboole.html#K10">|]</a></span> is   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="msualg_4.html#D3">MSUALG_4:def 3</a></i>;<br/>


</div>
