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<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">X</font> be  <a href="relat_1.html#V2">V3</a> <a href="finset_1.html#V2">V24</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1"/>
(  not <font color="Maroon">S</font> is <a href="msualg_1.html#V2">void</a> or <font color="Maroon">S</font> is <a href="msualg_1.html#V2">void</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><b>case </b></a><a NAME="E1:29_1_1"/>
not <font color="Maroon">S</font> is <a href="msualg_1.html#V2">void</a>
 ;<br/><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>assume </b><a NAME="E2:29_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">S'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S'</font>;<br/><b>assume </b><a NAME="E3:29_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font>
 ;<br/><b>reconsider </b><font color="Maroon">G</font> =  <a href="msafree.html#K13">FreeGen</a> <font color="Maroon">X</font> as    <a href="msafree.html#M1">GeneratorSet</a> of  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font> ;<br/><b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as    <a href="msafree.html#M1">GeneratorSet</a> of <font color="Maroon">A</font> <b>by </b><i>, </i>;<br/><b>take </b>
<font color="Maroon">G</font>
;<br/><b>thus </b><a NAME="E6:29_1_1"/>
<font color="Maroon">G</font> is <a href="finset_1.html#V2">locally-finite</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_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> ; <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="E1:29_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S'</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">iS</font> = <font color="Maroon">i</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Gi</font> = <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">Xi</font> = <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">iS</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="pre_circ.html#D3">PRE_CIRC:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1"><b>now </b></a><div class="add"><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><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span>;<br/><a NAME="E1:29_1_1_1_1"/><i><font color="Green">E33</font></i>: 
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> <font color="Maroon">Gi</font> holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xi</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1_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="E1:29_1_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font>
 ;<br/>

<a NAME="E2:29_1_1_1_1_1"/>
<font color="Maroon">Gi</font> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">iS</font>,<font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D18">MSAFREE:def 18</a></i>;<br/>
<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:29_1_1_1_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xi</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="msafree.html#D17">MSAFREE:def 17</a></i>;<br/>
<b>take </b>
<font color="Maroon">u</font>
;<br/>

<b>thus </b><a NAME="E5:29_1_1_1_1_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xi</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">e</font>,<font color="Maroon">u</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:29_1_1_1_1"/><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Gi</font>
 <b>and </b><br/><a NAME="E4:29_1_1_1_1"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Xi</font>
 <b>and </b><br/><a NAME="E5:29_1_1_1_1"/><i><font color="Green">E65</font></i>: 
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> <font color="Maroon">Gi</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i><b>take </b><font color="Maroon">f</font> = <font color="Maroon">f</font>;<br/><a NAME="E6:29_1_1_1_1"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1_2"><b>proof </b></a><div class="add">

<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><a NAME="E1:29_1_1_1_1_2"/><i><font color="Green">E66</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">f</font> &amp; <font color="Maroon">x2</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">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<b>hence </b><font color="Maroon">x1</font> = 
 <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">x2</font>
<b>by </b><i>, , </i>
;<br/><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:29_1_1_1_1"/>
 ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Gi</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Xi</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E6:29_1_1_1"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">Gi</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">Xi</font> or  <a href="card_1.html#K1">Card</a> <font color="Maroon">Gi</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">Xi</font> )
 
<b>by </b><i><a class="ref" href="card_1.html#T26">CARD_1:26</a></i>;<br/>
<b>hence </b><a NAME="E7:29_1_1_1"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="card_2.html#T68">CARD_2:68</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2"><b>case </b></a><a NAME="E1:29_1_2"/>
<font color="Maroon">S</font> is <a href="msualg_1.html#V2">void</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_1_2"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> is <a href="finset_1.html#V2">locally-finite</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
