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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:41"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_circ.html#V1">locally-finite</a>
 <b>and </b><br/><a NAME="E2:41"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R2">c=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="pboole.html#K10">|]</a></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> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">a<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">a<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> );<br/>
<a NAME="E3:41"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:41_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:41_1"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="mssubfam.html#E1:41"><i><font color="Green">E5</font></i></a>, <a class="ref" href="pre_circ.html#D3">PRE_CIRC:def 3</a></i>;<br/>
<a NAME="E3:41_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="mssubfam.html#E2:41"><i><font color="Green">E6</font></i></a>, <a class="txt" href="mssubfam.html#E1:41_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E4:41_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="mssubfam.html#E1:41_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:41_1"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="mssubfam.html#E2:41_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="cqc_the1.html#T13">CQC_THE1:13</a></i>;<br/>
<b>thus </b><a NAME="E7:41_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="mssubfam.html#E6:41_1"><i><font color="Green">E10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:41"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>(<a class="txt" href="mssubfam.html#E3:41"><i><font color="Green">E7</font></i></a>);<br/></i>
<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> ] means ( <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">a<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> );<br/>
<a NAME="E6:41"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<b>by </b><i><a class="txt" href="mssubfam.html#E5:41"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:41"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>(<a class="txt" href="mssubfam.html#E6:41"><i><font color="Green">E9</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E9:41"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_circ.html#V1">locally-finite</a>
 <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">c<sub>7</sub></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="E1:41_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:41_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="mssubfam.html#E8:41"><i><font color="Green">E10</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E10:41"/>
<font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <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">c<sub>7</sub></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><br/>

<b>assume </b><a NAME="E1:41_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:41_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="mssubfam.html#E8:41"><i><font color="Green">E10</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E11:41"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_circ.html#V1">locally-finite</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">c<sub>7</sub></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="E1:41_4"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:41_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="mssubfam.html#E8:41"><i><font color="Green">E10</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E12:41"/>
<font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></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><br/>

<b>assume </b><a NAME="E1:41_5"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:41_5"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="mssubfam.html#E8:41"><i><font color="Green">E10</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E13:41"/>
<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R2">c=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="pboole.html#K10">|]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></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><br/>

<b>assume </b><a NAME="E1:41_6"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:41_6"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="mssubfam.html#E8:41"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:41_6"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="mssubfam.html#E1:41_6"><i><font color="Green">E11</font></i></a>, <a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>;<br/>


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


</div>
