<?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">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">I</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">EqR</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">I</font>;<br/>







<b>set </b><font color="Maroon">U1</font> =  <a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">U2</font> =  <a href="pralg_2.html#K15">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">U1'</font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">U2'</font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span>;<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> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">f1</font>, <font color="Olive">g1</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>for <font color="Olive">e</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font> holds  <font color="Olive">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="relat_1.html#K7">|</a> <font color="Olive">e</font>;<br/>
<a NAME="E1:41"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/>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="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> 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="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>,<font color="Olive">s</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:41_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">s</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>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:41_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">s'</font> = <font color="Maroon">s</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E4:41_1"/><i><font color="Green">E44</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, </i>;<br/>
<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">x</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">y</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="E7:41_1"/><i><font color="Green">E45</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">y</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>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<a NAME="E8:41_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<i><font color="Green">E68</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> = 
 <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E2:41_1_1"/><i><font color="Green">E69</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:41_1_1"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> holds <br/><font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:41_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">A'</font> =  <a href="pralg_3.html#K6" target="_self">product</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span>;<br/>
<a NAME="E2:41_1_1_2"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="txt" href="pralg_3.html#E1:41"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:41_1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E5:41_1_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="txt" href="pralg_3.html#E1:41"><i><font color="Green">E38</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> <b>by </b><i>, <a class="txt" href="pralg_3.html#E1:41_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>
<i><font color="Green">E77</font></i>:  <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">a</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="subset_1.html#K9">/\</a> <font color="Maroon">a</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="subset_1.html#K9">/\</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">Aa1</font> = <font color="Maroon">A</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">a</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">a</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:41_1_1_2"/>
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">a</font> holds <br/><font color="Maroon">Aa1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_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="E1:41_1_1_2_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:41_1_1_2_2"/><b>then </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</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>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D12">PRALG_2:def 12</a></i>;<br/>
<b>hence </b><a NAME="E3:41_1_1_2_2"/>
<font color="Maroon">Aa1</font> <a href="funct_1.html#K1">.</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>
 <b>by </b><i>, <a class="txt" href="pralg_3.html#E2:41_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Aa</font> = <font color="Maroon">Aa1</font> as    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">a</font>,<font color="Maroon">S</font> <b>by </b><i><a class="ref" href="pralg_2.html#D12">PRALG_2:def 12</a></i>;<br/>
<a NAME="E11:41_1_1_2"/>
<font color="Maroon">x</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E12:41_1_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">Aa</font>,<font color="Maroon">s'</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Ji</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Cs</font> being    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">Ji</font>,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E14:41_1_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">Ji</font> <a href="hidden.html#R1">=</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">a</font>
 <b>and </b><br/><a NAME="E15:41_1_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">Cs</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E16:41_1_1_2"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">Cs</font>
 <b>by </b><i>, <a class="txt" href="pralg_3.html#E1:41"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E17:41_1_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">Ji</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="txt" href="pralg_3.html#E1:41"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pralg_3.html#E4:41_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<b>consider </b><font color="Maroon">U0</font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E19:41_1_1_2"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font> )
 <b>by </b><i>, <a class="txt" href="pralg_3.html#E1:41"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pralg_2.html#D16">PRALG_2:def 16</a></i>;<br/>
<a NAME="E20:41_1_1_2"/>
<span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">Aa</font>,<font color="Maroon">s'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_3.html#T2" target="_self">Th2</a>, , <a class="ref" href="pralg_3.html#T1" target="_self">Th1</a>, <a class="txt" href="pralg_3.html#E7:41_1"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<b>hence </b><a NAME="E21:41_1_1_2"/>
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="txt" href="pralg_3.html#E1:41"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pralg_3.html#E2:41_1"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:41_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:41_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">x</font>,<font color="Maroon">s</font>] )
 <b>by </b><i>, <a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being    <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:41"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/> ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</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="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</font>,<font color="Olive">s</font>] ) )
 <b>from </b><i><a class="ref" href="mssubfam.html#S1">MSSUBFAM:sch 1</a>();<br/></i>
<a NAME="E4:41"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">F</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font>, <a href="pralg_2.html#K15">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a><br/>

<b>assume </b><a NAME="E1:41_2"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span>;<br/>

<b>thus </b><a NAME="E2:41_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">s</font> =  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>;<br/>
<b>set </b><font color="Maroon">U3</font> =  <a href="pralg_3.html#K6" target="_self">product</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span><b> such that </b><br/><a NAME="E2:41_2_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 <b>and </b><br/><a NAME="E3:41_2_1"/><i><font color="Green">E145</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> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x'</font>,<font color="Olive">x'</font>, <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>]
 <b>by </b><i/>;<br/>
<a NAME="E4:41_2_1"/><i><font color="Green">E147</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<i><font color="Green">E148</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>
;<br/>
<a NAME="E6:41_2_1"/><i><font color="Green">E149</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span>
 
;<br/>
<a NAME="E7:41_2_1"/><b>then </b><i><font color="Green">E150</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>;<br/>
<a NAME="E8:41_2_1"/><i><font color="Green">E151</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_2_1_2"/>
(  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_2_1"><b>suppose </b></a><a NAME="E1:41_2_1_2_1"/><i><font color="Green">E152</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:41_2_1_2_1"/><b>then </b>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="pralg_2.html#T11">PRALG_2:11</a></i>;<br/><a NAME="E3:41_2_1_2_1"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:41_2_1_2_1"/><b>then </b>
 <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:41"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E5:41_2_1_2_1"/><b>then </b><i><font color="Green">E154</font></i>: 
 <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>;<br/><a NAME="E6:41_2_1_2_1"/><i><font color="Green">E155</font></i>: 
 <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><b>then </b><i><font color="Green">E156</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><a NAME="E8:41_2_1_2_1"/><i><font color="Green">E157</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">g1</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, </i>;<br/><i><font color="Green">E158</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="pralg_3.html#T2" target="_self">Th2</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
 <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><a NAME="E11:41_2_1_2_1"/>
 <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font></span>)</span>,<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p0">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K6">.</a> <font color="Olive">i'</font></span>)</span></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font> : verum </span>}</span> </span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E12:41_2_1_2_1"/><b>then </b><i><font color="Green">E159</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T169">FUNCT_2:169</a></i>;<br/><div><i><font color="Green">E160</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_2_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>;<br/><b>consider </b><font color="Maroon">Ji</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Cs</font> being    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">Ji</font>,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E2:41_2_1_2_1_3"/><i><font color="Green">E161</font></i>: 
( <font color="Maroon">Ji</font> <a href="hidden.html#R1">=</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">e</font> &amp; <font color="Maroon">Cs</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K6">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">Cs</font> )
 <b>by </b><i/>;<br/><i><font color="Green">E162</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font></span>)</span> = 
<font color="Maroon">I</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E4:41_2_1_2_1_3"/>
 <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">Cs</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">Ji</font>,<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<span class="p4">(<span class="default"><font color="Maroon">Cs</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i'</font></span>)</span></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ji</font> : verum </span>}</span> </span>)</span>
 <b>by </b><i>, </i>;<br/><b>then </b><i><font color="Green">E163</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">Cs</font></span>)</span></span>)</span> = 
<font color="Maroon">Ji</font>
<b>by </b><i><a class="ref" href="funct_2.html#T169">FUNCT_2:169</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_2_1_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:41_2_1_2_1_3_3"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e</font>
 ;<br/><b>then reconsider </b><font color="Maroon">ii</font> = <font color="Maroon">i1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ji</font> <b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>reconsider </b><font color="Maroon">ii'</font> = <font color="Maroon">ii</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> <b>by </b><i/>;<br/><i><font color="Green">E166</font></i>:  <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">e</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="subset_1.html#K9">/\</a> <font color="Maroon">e</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="subset_1.html#K9">/\</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E5:41_2_1_2_1_3_3"/>
<font color="Maroon">Cs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:41_2_1_2_1_3_3"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">Cs</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">ii</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">ii'</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> = 
<span class="p1">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
 <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">ii'</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">Cs</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
<b>by </b><i>, , </i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:41_2_1_2_1_3"/>
<span class="p1">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K6">.</a> <font color="Maroon">e</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="pralg_3.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_2_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:41_2_1_2_1_4"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>
 ;<br/><b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font> ;<br/><b>consider </b><font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:41_2_1_2_1_4"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><a NAME="E5:41_2_1_2_1_4"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> = 
 <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K6">.</a> <font color="Maroon">e</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><a NAME="E15:41_2_1_2_1"/><b>then </b><i><font color="Green">E169</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>thus </b><a NAME="E16:41_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_2_2"><b>suppose </b></a><a NAME="E1:41_2_1_2_2"/><i><font color="Green">E170</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:41_2_1_2_2"/><i><font color="Green">E171</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">g1</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, </i>;<br/><a NAME="E4:41_2_1_2_2"/><i><font color="Green">E172</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">f1</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i/>;<br/><i><font color="Green">E173</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
 <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><i><font color="Green">E174</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
 <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><div><i><font color="Green">E175</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_2_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>assume </b><a NAME="E1:41_2_1_2_2_3"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/><b>consider </b><font color="Maroon">Ji</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Cs</font> being    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">Ji</font>,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:41_2_1_2_2_3"/><i><font color="Green">E177</font></i>: 
( <font color="Maroon">Ji</font> <a href="hidden.html#R1">=</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">e</font> &amp; <font color="Maroon">Cs</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K6">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">Cs</font> )
 <b>by </b><i/>;<br/> <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><b>then </b><i><font color="Green">E178</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font></span>)</span> = 
<font color="Maroon">I</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E6:41_2_1_2_2_3"/><i><font color="Green">E179</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K6">.</a> <font color="Maroon">e</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E7:41_2_1_2_2_3"/><i><font color="Green">E180</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">Cs</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E8:41_2_1_2_2_3"/><i><font color="Green">E181</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K6">.</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">Cs</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>then reconsider </b><font color="Maroon">f3</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K6">.</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i/>;<br/><i><font color="Green">E182</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f3</font> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">Cs</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#T3" target="_self">Th3</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon">Ji</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/><div><i><font color="Green">E183</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_2_2_3_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><b>assume </b><a NAME="E1:41_2_1_2_2_3_4"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i2</font> = <font color="Maroon">i1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ji</font> <b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E3:41_2_1_2_2_3_4"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">Cs</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i2</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/><a NAME="E4:41_2_1_2_2_3_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">Cs</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i2</font></span>)</span>
 ;<br/><a NAME="E5:41_2_1_2_2_3_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">Cs</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i2</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>;<br/><b>then </b><i><font color="Green">E209</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">Cs</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i2</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>
;<br/><a NAME="E7:41_2_1_2_2_3_4"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E8:41_2_1_2_2_3_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span>
 ;<br/><a NAME="E9:41_2_1_2_2_3_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>;<br/><b>then </b><i><font color="Green">E210</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_2_2_3_4_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:41_2_1_2_2_3_4_3"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 ;<br/><a NAME="E2:41_2_1_2_2_3_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">s1</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ;<br/><a NAME="E4:41_2_1_2_2_3_4_3"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:41_2_1_2_2_3_4_3"/><b>then </b><i><font color="Green">E213</font></i>: 
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">f4</font> = <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, </i>;<br/><a NAME="E7:41_2_1_2_2_3_4_3"/>
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
 <b>by </b><i>, <a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/><a NAME="E8:41_2_1_2_2_3_4_3"/><b>then </b><i><font color="Green">E215</font></i>: 
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
 ;<br/> <a href="relat_1.html#K1">dom</a> <font color="Maroon">f4</font> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><b>then </b><i><font color="Green">E216</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f4</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font></span>)</span> = 
<font color="Maroon">I</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E11:41_2_1_2_2_3_4_3"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">f5</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">f7</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font><b> such that </b><br/><a NAME="E14:41_2_1_2_2_3_4_3"/><i><font color="Green">E219</font></i>: 
<font color="Maroon">f7</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s1</font>
 <b>and </b><br/><a NAME="E15:41_2_1_2_2_3_4_3"/><i><font color="Green">E220</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> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f7</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x'</font>,<font color="Olive">x'</font>,<font color="Maroon">s1</font>]
 <b>by </b><i/>;<br/><font color="Maroon">f5</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f7</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><a NAME="E17:41_2_1_2_2_3_4_3"/><b>then </b><i><font color="Green">E221</font></i>: 
<font color="Maroon">f5</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f4</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">f6</font> = <font color="Maroon">f5</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><a NAME="E19:41_2_1_2_2_3_4_3"/>
<font color="Maroon">f6</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f4</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font>
<b>by </b><i>, , , , </i>
<br/>.= 
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E12:41_2_1_2_2_3_4"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="txt" href="pralg_3.html#E4"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_2_2_3_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:41_2_1_2_2_3_5"/><i><font color="Green">E223</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i2</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Ji</font> <b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i2</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> <b>by </b><i/>;<br/><i><font color="Green">E224</font></i>:  <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">e</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="subset_1.html#K9">/\</a> <font color="Maroon">e</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="subset_1.html#K9">/\</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E5:41_2_1_2_2_3_5"/>
<font color="Maroon">Cs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:41_2_1_2_2_3_5"/><b>then </b>
<font color="Maroon">Cs</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E13:41_2_1_2_2_3"/>
<font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K6">.</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_2_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:41_2_1_2_2_4"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>
 ;<br/><b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font> ;<br/><b>reconsider </b><font color="Maroon">f2</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, </i>;<br/><a NAME="E4:41_2_1_2_2_4"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font>
 <b>by </b><i>, , <a class="ref" href="pralg_3.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K6">.</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E10:41_2_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div><b>end;</b></div>
<a NAME="E5:41"/><i><font color="Green">E225</font></i>: 
<font color="Maroon">F</font> is <a href="msualg_3.html#V2">"onto"</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a><br/>

<b>assume </b><a NAME="E1:41_3"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">s</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>reconsider </b><font color="Maroon">s'</font> = <font color="Maroon">s</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E3:41_3"/><i><font color="Green">E227</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<a NAME="E4:41_3"/><i><font color="Green">E228</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> 
 
;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E6:41_3"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E7:41_3"/><i><font color="Green">E230</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> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</font>,<font color="Maroon">s</font>]
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:41_3"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> iff  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:41_3_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> iff  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:41_3_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> implies  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_3_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E2:41_3_1_1_1"/><b>then </b><i><font color="Green">E231</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<b>then consider </b><font color="Maroon">gg</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:41_3_1_1_1"/><i><font color="Green">E232</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gg</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">gg</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> )
 <b>and </b><br/><a NAME="E5:41_3_1_1_1"/><i><font color="Green">E233</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="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> holds <br/><font color="Maroon">gg</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x'</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, </i>;<br/>
<a NAME="E7:41_3_1_1_1"/><i><font color="Green">E234</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">e</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font> ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">e</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E8:41_3_1_1_1"/><i><font color="Green">E236</font></i>: 
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/> ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1_1_1_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="E1:41_3_1_1_1_1"/><i><font color="Green">E237</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:41_3_1_1_1_1"/><i><font color="Green">E238</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:41_3_1_1_1_1"/>
 <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">e</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font><b> such that </b><br/><a NAME="E5:41_3_1_1_1_1"/><i><font color="Green">E239</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">i</font>
 ;<br/>
<a NAME="E6:41_3_1_1_1_1"/><i><font color="Green">E240</font></i>: 
<font color="Maroon">gg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="pralg_3.html#T2" target="_self">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon">Ji</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Cs</font> being    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">Ji</font>,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E8:41_3_1_1_1_1"/>
<font color="Maroon">Ji</font> <a href="hidden.html#R1">=</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">e</font>
 <b>and </b><br/><a NAME="E9:41_3_1_1_1_1"/>
<font color="Maroon">Cs</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>and </b><br/><a NAME="E10:41_3_1_1_1_1"/><i><font color="Green">E241</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K6">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">Cs</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">U0</font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E12:41_3_1_1_1_1"/><i><font color="Green">E242</font></i>: 
( <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K6">.</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font> )
 <b>by </b><i><a class="ref" href="pralg_2.html#D16">PRALG_2:def 16</a></i>;<br/>
<a NAME="E13:41_3_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">Cs</font>,<font color="Maroon">s'</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, , </i>;<br/>
<a NAME="E15:41_3_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">y'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">y'</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E2:41_3_1_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> &amp; <font color="Maroon">y'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:41_3_1_1_1_1"/>
 ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>,<font color="Olive">j</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">x</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E10:41_3_1_1_1"/><i><font color="Green">E244</font></i>: 
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/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<a NAME="E11:41_3_1_1_1"/><i><font color="Green">E245</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1_1_1_2"><b>proof </b></a><div class="add">

<i><font color="Green">E246</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x</font> = 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E2:41_3_1_1_1_2"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span> holds <br/><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1_1_1_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:41_3_1_1_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span>
 ;<br/>

<a NAME="E2:41_3_1_1_1_2_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">e</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>, <font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:41_3_1_1_1_2_2"/><i><font color="Green">E248</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e</font> &amp; <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> )
 <b>and </b><br/><a NAME="E5:41_3_1_1_1_2_2"/><i><font color="Green">E249</font></i>: 
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:41_3_1_1_1_2_2"/><i><font color="Green">E250</font></i>: 
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 
<b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">Ji</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Cs</font> being    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">Ji</font>,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E8:41_3_1_1_1_2_2"/><i><font color="Green">E251</font></i>: 
<font color="Maroon">Ji</font> <a href="hidden.html#R1">=</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">e</font>
 <b>and </b><br/><a NAME="E9:41_3_1_1_1_2_2"/><i><font color="Green">E252</font></i>: 
<font color="Maroon">Cs</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>and </b><br/><a NAME="E10:41_3_1_1_1_2_2"/><i><font color="Green">E253</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K6">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">Cs</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">U0</font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E12:41_3_1_1_1_2_2"/><i><font color="Green">E254</font></i>: 
( <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K6">.</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font> )
 <b>by </b><i><a class="ref" href="pralg_2.html#D16">PRALG_2:def 16</a></i>;<br/>
<a NAME="E13:41_3_1_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">Cs</font>,<font color="Maroon">s'</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E15:41_3_1_1_1_2_2"/><i><font color="Green">E255</font></i>: 
( <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">Cs</font>,<font color="Maroon">s'</font></span>)</span> )
 <b>and </b><br/><a NAME="E16:41_3_1_1_1_2_2"/><i><font color="Green">E256</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="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">Cs</font>,<font color="Maroon">s'</font></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">Cs</font>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x'</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">e</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="subset_1.html#K9">/\</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E18:41_3_1_1_1_2_2"/><b>then </b><i><font color="Green">E257</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E19:41_3_1_1_1_2_2"/><i><font color="Green">E258</font></i>: 
<font color="Maroon">Cs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="pralg_3.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E20:41_3_1_1_1_2_2"/>
<font color="Maroon">Ji</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<a NAME="E21:41_3_1_1_1_2_2"/><b>then </b><i><font color="Green">E259</font></i>: 
 <a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">Cs</font>,<font color="Maroon">s'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font>
 
<b>by </b><i>, </i>;<br/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">Cs</font>,<font color="Maroon">s'</font></span>)</span> = 
<font color="Maroon">Ji</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>
<b>hence </b><a NAME="E23:41_3_1_1_1_2_2"/>
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, , , , <a class="txt" href="pralg_3.html#E4"><i><font color="Green">Lemma57</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:41_3_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font>
 <b>by </b><i>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:41_3_1_1_1"/><b>then </b><i><font color="Green">E260</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:41_3_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E14:41_3_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
;<br/>
<a NAME="E15:41_3_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<b>then consider </b><font color="Maroon">gg'</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E17:41_3_1_1_1"/><i><font color="Green">E262</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gg'</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">gg'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> )
 <b>and </b><br/><a NAME="E18:41_3_1_1_1"/>
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="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> holds <br/><font color="Maroon">gg'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x'</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i/>;<br/>
<a NAME="E20:41_3_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:41_3_1_1_1_3"/>
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</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">y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1_1_1_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:41_3_1_1_1_3_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font> ;<br/>
<a NAME="E3:41_3_1_1_1_3_1"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 
<b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">Ji</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Cs</font> being    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">Ji</font>,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E5:41_3_1_1_1_3_1"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">Ji</font> <a href="hidden.html#R1">=</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">e</font>
 <b>and </b><br/><a NAME="E6:41_3_1_1_1_3_1"/>
<font color="Maroon">Cs</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>and </b><br/><a NAME="E7:41_3_1_1_1_3_1"/><i><font color="Green">E265</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K6">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">Cs</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">U0</font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E9:41_3_1_1_1_3_1"/><i><font color="Green">E266</font></i>: 
( <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K6">.</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font> )
 <b>by </b><i><a class="ref" href="pralg_2.html#D16">PRALG_2:def 16</a></i>;<br/>
<a NAME="E10:41_3_1_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">Cs</font>,<font color="Maroon">s'</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pralg_3.html#T3" target="_self">Th3</a>, <a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E12:41_3_1_1_1_3_1"/><i><font color="Green">E267</font></i>: 
( <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">Cs</font>,<font color="Maroon">s'</font></span>)</span> )
 <b>and </b><br/><a NAME="E13:41_3_1_1_1_3_1"/>
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="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">Cs</font>,<font color="Maroon">s'</font></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">Cs</font>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x'</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i/>;<br/>
<a NAME="E15:41_3_1_1_1_3_1"/>
<font color="Maroon">x</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1_1_1_3_1_1"><b>proof </b></a><div class="add">

<i><font color="Green">E268</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">e</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="subset_1.html#K9">/\</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<i><font color="Green">E269</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">y'</font> = 
<font color="Maroon">Ji</font>
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>
<a NAME="E3:41_3_1_1_1_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> <font color="Maroon">e</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1_1_1_3_1_1_3"><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="E1:41_3_1_1_1_3_1_1_3"/><i><font color="Green">E270</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e</font>
 ;<br/>

<b>then consider </b><font color="Maroon">e1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>, <font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:41_3_1_1_1_3_1_1_3"/><i><font color="Green">E271</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e1</font>
 <b>and </b><br/><a NAME="E4:41_3_1_1_1_3_1_1_3"/><i><font color="Green">E272</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e1</font>
 <b>and </b><br/><a NAME="E5:41_3_1_1_1_3_1_1_3"/><i><font color="Green">E273</font></i>: 
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:41_3_1_1_1_3_1_1_3"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font>
 
<b>by </b><i>, <a class="txt" href="pralg_3.html#E4"><i><font color="Green">Lemma57</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E7:41_3_1_1_1_3_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:41_3_1_1_1_3_1_1"/>
<font color="Maroon">x</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:41_3_1_1_1_3_1"/>
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="pralg_3.html#T1" target="_self">Th1</a>, <a class="ref" href="pralg_3.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:41_3_1_1_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E21:41_3_1_1_1"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> )
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:41_3_1_1"/>
(  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1_1_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:41_3_1_1_2"/><i><font color="Green">E274</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:41_3_1_1_2"/><i><font color="Green">E275</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E3:41_3_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:41_3_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/>


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


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:41_3"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:41"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:41"/><i><font color="Green">E276</font></i>: 
<font color="Maroon">F</font> is <a href="msualg_3.html#V1">"1-1"</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D2">MSUALG_3:def 2</a><br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E1:41_4"/><i><font color="Green">E277</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 ;<br/>

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:41_4"/><i><font color="Green">E278</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E3:41_4"/><i><font color="Green">E279</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:41_4"/><i><font color="Green">E280</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/>

<a NAME="E5:41_4"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E7:41_4"/><i><font color="Green">E281</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E8:41_4"/><i><font color="Green">E282</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> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</font>,<font color="Maroon">s</font>]
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:41_4"/><i><font color="Green">E283</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E10:41_4"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">s'</font> = <font color="Maroon">s</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E2:41_4_1"/><i><font color="Green">E284</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<a NAME="E3:41_4_1"/><i><font color="Green">E285</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>,<font color="Maroon">s'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<a NAME="E4:41_4_1"/><i><font color="Green">E286</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> 
 
;<br/>
<b>consider </b><font color="Maroon">gg</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:41_4_1"/><i><font color="Green">E287</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gg</font>
 <b>and </b><br/><a NAME="E7:41_4_1"/><i><font color="Green">E288</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">gg</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span>
 <b>and </b><br/><a NAME="E8:41_4_1"/>
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="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span> holds <br/><font color="Maroon">gg</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x'</font>
 <b>by </b><i>, , , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="pralg_3.html#T2" target="_self">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon">gg1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:41_4_1"/><i><font color="Green">E290</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gg1</font>
 <b>and </b><br/><a NAME="E12:41_4_1"/><i><font color="Green">E291</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">gg1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span>
 <b>and </b><br/><a NAME="E13:41_4_1"/>
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="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span> holds <br/><font color="Maroon">gg1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x'</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:41"><i><font color="Green">E38</font></i></a>, , , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x2</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i/>;<br/>
<a NAME="E15:41_4_1"/><i><font color="Green">E292</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pralg_3.html#T2" target="_self">Th2</a>, , <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E16:41_4_1"/><i><font color="Green">E293</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i>, , <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E17:41_4_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> <font color="Maroon">I</font> holds <br/><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4_1_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="E1:41_4_1_1"/><i><font color="Green">E294</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:41_4_1_1"/><b>then </b><i><font color="Green">E295</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>
<a NAME="E3:41_4_1_1"/><i><font color="Green">E296</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EqR</font>
 
<b>by </b><i><a class="ref" href="pralg_3.html#T3" target="_self">Th3</a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E4:41_4_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="txt" href="pralg_3.html#E1:41"><i><font color="Green">E38</font></i></a>, , , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f''</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, </i>;<br/>
<font color="Maroon">x1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">i</font></span>)</span> = 
<font color="Maroon">f''</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="pralg_3.html#T1" target="_self">Th1</a>, , , , </i>
<br/>.= 
<font color="Maroon">x2</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_3.html#E1:41"><i><font color="Green">E38</font></i></a>, , , </i>
;<br/>
<b>then </b><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EqR</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><a NAME="E8:41_4_1_1"/>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:41_4_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E7:41"/>
<font color="Maroon">F</font> <a href="msualg_3.html#R4">is_isomorphism</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font>, <a href="pralg_2.html#K15">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:41_5"/>
<font color="Maroon">F</font> <a href="msualg_3.html#R2">is_epimorphism</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font>, <a href="pralg_2.html#K15">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D12">MSUALG_3:def 12</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_5_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:41_5_1"/>
( <font color="Maroon">F</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font>, <a href="pralg_2.html#K15">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> &amp; <font color="Maroon">F</font> is <a href="msualg_3.html#V2">"onto"</a> )
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a><br/>


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

<b>thus </b><a NAME="E2:41_5"/>
<font color="Maroon">F</font> <a href="msualg_3.html#R3">is_monomorphism</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font>, <a href="pralg_2.html#K15">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_5_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:41_5_2"/>
( <font color="Maroon">F</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font>, <a href="pralg_2.html#K15">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> &amp; <font color="Maroon">F</font> is <a href="msualg_3.html#V1">"1-1"</a> )
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D11">MSUALG_3:def 11</a><br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:41"/>
 <a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font>, <a href="pralg_2.html#K15">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_3.html#K5" target="_self">/</a> <font color="Maroon">EqR</font></span>)</span></span>)</span> <a href="msualg_3.html#R6">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="msualg_3.html#D13">MSUALG_3:def 13</a></i>;<br/>


</div>
