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

<b>thus </b><a NAME="E1:39_1_1"/>
 ex <font color="Olive">X</font> being   <a href="pralg_3.html#M1" target="_self">MSAlgebra-Class</a> of <font color="Maroon">S</font>, <a href="pralg_3.html#K1" target="_self">id</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font></span>)</span> st <br/>for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font> holds <br/><font color="Olive">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="relat_1.html#K7">|</a> <font color="Olive">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">A</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font><b> such that </b><br/><a NAME="E2:39_1_1_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font> holds <br/><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">c</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S4">PBOOLE:sch 4</a>();<br/></i>
<a NAME="E3:39_1_1_1"/>
<font color="Maroon">X</font> is    <a href="pralg_3.html#M1" target="_self">MSAlgebra-Class</a> of <font color="Maroon">S</font>, <a href="pralg_3.html#K1" target="_self">id</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pralg_3.html#D4" target="_self">PRALG_3:def 4</a><br/>

<b>assume </b><a NAME="E1:39_1_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>
 ;<br/>

<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_1"><b>now </b></a><div class="add"><b>thus </b> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K1" target="_self">id</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">ac</font> = <font color="Maroon">A</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1">(<span class="default"><a href="pralg_3.html#K1" target="_self">id</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:39_1_1_1_1"/>
<font color="Maroon">ac</font> is    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <span class="p1">(<span class="default"><a href="pralg_3.html#K1" target="_self">id</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pralg_2.html#D12">PRALG_2:def 12</a><br/>

<b>assume </b><a NAME="E1:39_1_1_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K1" target="_self">id</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
 ;<br/>

<a NAME="E2:39_1_1_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">ac</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> <b>by </b><i>, , </i>;<br/>
<a NAME="E4:39_1_1_1_1_2"/>
<font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i'</font> is  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>
 
;<br/>
<b>hence </b><a NAME="E5:39_1_1_1_1_2"/>
<font color="Maroon">ac</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:39_1_1_1_1"/>
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> is    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <span class="p1">(<span class="default"><a href="pralg_3.html#K1" target="_self">id</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">S</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as    <a href="pralg_3.html#M1" target="_self">MSAlgebra-Class</a> of <font color="Maroon">S</font>, <a href="pralg_3.html#K1" target="_self">id</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>thus </b><a NAME="E5:39_1_1_1"/>
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font> holds <br/><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="relat_1.html#K7">|</a> <font color="Olive">c</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>


</div>
