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

<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">I</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">I</font>,<font color="Maroon">S</font>;<br/>







<b>assume </b><a NAME="E1:32"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>  ex <font color="Olive">C</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font> st <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font>
 ;<br/>

<b>set </b><font color="Maroon">J</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/>
<b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">A</font> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">D</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font> &amp; <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/>
<b>consider </b><font color="Maroon">W</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:32"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:32"/>
<font color="Maroon">W</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/>
<a NAME="E6:32"/><b>then </b>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">A</font> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">D</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font> &amp; <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">A</font> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">D</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font> &amp; <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">Q</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font> ex <font color="Olive">G</font> being   <a href="msafree.html#M1">GeneratorSet</a> of <font color="Olive">Q</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> &amp; <font color="Olive">G</font> is <a href="relat_1.html#V2">non-empty</a> &amp; <font color="Olive">G</font> is <a href="finset_1.html#V2">locally-finite</a> );<br/>
<a NAME="E8:32"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>  ex <font color="Olive">b</font> being   <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>;<br/>

<a NAME="E1:32_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
;<br/>
<b>then consider </b><font color="Maroon">C</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:32_1"/><i><font color="Green">E41</font></i>: 
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">D</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font> st <br/>( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> )
 ;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Maroon">D</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E6:32_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">G</font> being    <a href="msafree.html#M1">GeneratorSet</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E8:32_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">G</font> is <a href="finset_1.html#V2">locally-finite</a>
 <b>by </b><i><a class="ref" href="msafree2.html#D10">MSAFREE2:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">S</font> being  <a href="relat_1.html#V2">V2</a> <a href="finset_1.html#V2">V19</a>  <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E10:32_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">G</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="msualg_9.html#T8">MSUALG_9:8</a></i>;<br/>
<a NAME="E11:32_1"/>
<font color="Maroon">S</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">S</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E2:32_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E3:32_1_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>hence </b><a NAME="E4:32_1_1"/>
<font color="Maroon">S</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#T15">PBOOLE:15</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">S</font> as    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="closure2.html#D1">CLOSURE2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

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

<b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S</font> as    <a href="msafree.html#M1">GeneratorSet</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="msscyc_1.html#T21">MSSCYC_1:21</a></i>;<br/>
<b>take </b>
<font color="Maroon">S</font>
;<br/>

<b>thus </b><a NAME="E14:32_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E15:32_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> &amp; <font color="Maroon">S</font> is <a href="relat_1.html#V2">non-empty</a> &amp; <font color="Maroon">S</font> is <a href="finset_1.html#V2">locally-finite</a> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">Z</font>, <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E10:32"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">B</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<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>  =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">SOR</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E12:32"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">j</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/><font color="Maroon">SOR</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S4">PBOOLE:sch 4</a>();<br/></i>
<a NAME="E13:32"/>
<font color="Maroon">SOR</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a>,<a class="ref" href="pboole.html#D23">PBOOLE:def 23</a><br/>

<b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">j</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>set </b><font color="Maroon">UU</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span> ;<br/>
<b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:32_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SOR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E3:32_2"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:32_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E6:32_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span><b> such that </b><br/><a NAME="E8:32_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E9:32_2"/>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E10:32_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">SOR</font> = <font color="Maroon">SOR</font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">A</font> ;<br/>
<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:32_3"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">j</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>set </b><font color="Maroon">UU</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span> ;<br/><b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><b>consider </b><font color="Maroon">C</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:32_3"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/><a NAME="E5:32_3"/>
<font color="Maroon">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/><a NAME="E6:32_3"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">Q</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E8:32_3"/><i><font color="Green">E117</font></i>: 
 ex <font color="Olive">G</font> being   <a href="msafree.html#M1">GeneratorSet</a> of <font color="Maroon">Q</font> st <br/>( <font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> &amp; <font color="Olive">G</font> is <a href="relat_1.html#V2">non-empty</a> &amp; <font color="Olive">G</font> is <a href="finset_1.html#V2">locally-finite</a> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">G1</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> as  <a href="finset_1.html#V2">V19</a>  <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <b>by </b><i>, , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E10:32_3"/>
<font color="Maroon">G1</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E11:32_3"/><b>then </b>
<font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><a NAME="E12:32_3"/><b>then </b>
<font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span> 
 ;<br/><b>hence </b><a NAME="E13:32_3"/>
not <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/></div><b>end;</b></div>
<a NAME="E16:32"/>
<font color="Maroon">SOR</font> is <a href="relat_1.html#V2">non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a><br/>

<b>assume </b><a NAME="E1:32_4"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">j</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>set </b><font color="Maroon">UU</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/>
<b>consider </b><font color="Maroon">C</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:32_4"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:32_4"/>
<font color="Maroon">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/>
<a NAME="E6:32_4"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">Q</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E8:32_4"/><i><font color="Green">E132</font></i>: 
 ex <font color="Olive">G</font> being   <a href="msafree.html#M1">GeneratorSet</a> of <font color="Maroon">Q</font> st <br/>( <font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> &amp; <font color="Olive">G</font> is <a href="relat_1.html#V2">non-empty</a> &amp; <font color="Olive">G</font> is <a href="finset_1.html#V2">locally-finite</a> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">G1</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> as  <a href="finset_1.html#V2">V19</a>  <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <b>by </b><i>, , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E10:32_4"/>
<font color="Maroon">G1</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E11:32_4"/><b>then </b>
<font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E12:32_4"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span> 
 
;<br/>
<a NAME="E13:32_4"/>
not <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, , <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<b>then consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:32_4"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E16:32_4"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span> 
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E17:32_4"/>
not <font color="Maroon">SOR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">SOR</font> = <font color="Maroon">SOR</font> as  <a href="relat_1.html#V2">V2</a> <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">A</font> ;<br/>
<a NAME="E18:32"/>
<font color="Maroon">SOR</font> is <a href="finset_1.html#V2">locally-finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_5"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">j</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>set </b><font color="Maroon">UU</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon">VV</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E137</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">r</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E2:32_5_1"/><i><font color="Green">E138</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T73">FINSEQ_1:73</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">L</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">L</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> );<br/><a NAME="E3:32_5_1"/><i><font color="Green">E140</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">VV</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/>

<b>consider </b><font color="Maroon">W</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E2:32_5_1_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>
<a NAME="E3:32_5_1_1"/>
<font color="Maroon">W</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/>
<a NAME="E4:32_5_1_1"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">Q</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E6:32_5_1_1"/><i><font color="Green">E143</font></i>: 
 ex <font color="Olive">G</font> being   <a href="msafree.html#M1">GeneratorSet</a> of <font color="Maroon">Q</font> st <br/>( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> &amp; <font color="Olive">G</font> is <a href="relat_1.html#V2">non-empty</a> &amp; <font color="Olive">G</font> is <a href="finset_1.html#V2">locally-finite</a> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">G1</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">a</font></span>)</span> as  <a href="finset_1.html#V2">V19</a>  <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <b>by </b><i>, , <a class="ref" href="equation.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E8:32_5_1_1"/>
<font color="Maroon">G1</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E9:32_5_1_1"/><b>then </b>
<font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E10:32_5_1_1"/><b>then </b>
<font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">VV</font> ;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>take </b>
<font color="Maroon">G1</font>
;<br/>

<b>thus </b><a NAME="E12:32_5_1_1"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I</font>,<font color="Maroon">VV</font><b> such that </b><br/><a NAME="E5:32_5_1"/><i><font color="Green">E144</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i><a NAME="E6:32_5_1"/><i><font color="Green">E145</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">r</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#T20">FINSEQ_1:20</a></i>;<br/><b>take </b><font color="Maroon">p</font> = <font color="Maroon">p</font>;<br/><b>thus </b><a NAME="E8:32_5_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">VV</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_5_1_2"/><i><font color="Green">E146</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_5_1_2_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 ;<br/>

<b>hence </b><a NAME="E2:32_5_1_2_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T25">FUNCT_1:25</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:32_5_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">VV</font>
 <b>by </b><i><a class="ref" href="equation.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E3:32_5_1_2"/>
<font color="Maroon">VV</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5_1_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_5_1_2_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">VV</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span><b> such that </b><br/><a NAME="E3:32_5_1_2_2"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:32_5_1_2_2"/><i><font color="Green">E148</font></i>: 
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 ;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Maroon">K</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E6:32_5_1_2_2"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:32_5_1_2_2"/><i><font color="Green">E150</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/>
<a NAME="E8:32_5_1_2_2"/><i><font color="Green">E151</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:32_5_1_2_2"/>
 ex <font color="Olive">L</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">L</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:32_5_1_2_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:32_5"/>
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> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span>  holds <br/><font color="Olive">x</font> is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_5_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span><b> such that </b><br/><a NAME="E3:32_5_2"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:32_5_2"/><i><font color="Green">E153</font></i>: 
 ex <font color="Olive">w</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">w</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 ;<br/>
<b>consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Maroon">K</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E6:32_5_2"/><i><font color="Green">E154</font></i>: 
( <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">w</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">E</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E8:32_5_2"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">w</font>
 <b>by </b><i/>;<br/>
<a NAME="E9:32_5_2"/>
<font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">w</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/>
<a NAME="E10:32_5_2"/><b>then </b>
<font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:32_5_2"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">w</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">w</font></span>)</span>]
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">Q</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E13:32_5_2"/>
<font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E14:32_5_2"/><i><font color="Green">E156</font></i>: 
 ex <font color="Olive">G</font> being   <a href="msafree.html#M1">GeneratorSet</a> of <font color="Maroon">Q</font> st <br/>( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> &amp; <font color="Olive">G</font> is <a href="finset_1.html#V2">locally-finite</a> )
 ;<br/>
<b>thus </b><a NAME="E15:32_5_2"/>
<font color="Maroon">x</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, , , <a class="ref" href="equation.html#T1" target="_self">Th1</a>, <a class="ref" href="pre_circ.html#D3">PRE_CIRC:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:32_5"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/>
<b>hence </b><a NAME="E6:32_5"/>
<font color="Maroon">SOR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">SOR</font> = <font color="Maroon">SOR</font> as  <a href="relat_1.html#V2">V2</a> <a href="finset_1.html#V2">V19</a> <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">A</font> ;<br/>
<b>take </b><font color="Maroon">B</font> =  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">SOR</font>;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/>

<b>consider </b><font color="Maroon">C</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E21:32"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/>
<a NAME="E22:32"/>
<font color="Maroon">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/>
<a NAME="E23:32"/><b>then </b>
<font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">Q</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font>, <font color="Maroon">G</font> being    <a href="msafree.html#M1">GeneratorSet</a> of <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E25:32"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E26:32"/><i><font color="Green">E159</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> is <a href="relat_1.html#V2">non-empty</a> &amp; <font color="Maroon">G</font> is <a href="finset_1.html#V2">locally-finite</a> )
 <b>by </b><i/>;<br/>
<a NAME="E27:32"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">G</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E28:32"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">Q</font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E29:32"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">Q</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E30:32"/><b>then </b><i><font color="Green">E161</font></i>: 
<font color="Maroon">G</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#T15">PBOOLE:15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G1</font> = <font color="Maroon">G</font> as  <a href="relat_1.html#V2">V2</a> <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E32:32"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">G1</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">SOR</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a>,<a class="ref" href="pboole.html#D23">PBOOLE:def 23</a><br/>

<b>assume </b><a NAME="E1:32_6"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">j</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>set </b><font color="Maroon">UU</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span> ;<br/>
<b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:32_6"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E3:32_6"/>
<font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="equation.html#T1" target="_self">Th1</a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E4:32_6"/><b>then </b>
<font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:32_6"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> :  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ex <font color="Olive">K</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:32_6"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SOR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="equation.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<font color="Maroon">C</font> = 
 <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">G</font>
<b>by </b><i>, , <a class="ref" href="msafree.html#T3">MSAFREE:3</a></i>
<br/>.= 
 <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="extens_1.html#T22">EXTENS_1:22</a></i>
;<br/>
<b>hence </b><a NAME="E34:32"/>
<font color="Maroon">F</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">B</font>
 <b>by </b><i>, , <a class="ref" href="extens_1.html#T21">EXTENS_1:21</a></i>;<br/>


</div>
