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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="coh_sp.html#NM1">Coherence_Space</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:92"/><i><font color="Green">E49</font></i>: 
for <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 <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being <a href="cohsp_1.html#V9" target="_self">U-stable</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K8" target="_self">Trace</a> <font color="Olive">b<sub>3</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E2:92"/><i><font color="Green">E50</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</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>1</sub></font>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_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> ; <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="E1:92_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:92_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:92_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="cohsp_1.html#V9" target="_self">U-stable</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:92_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K8" target="_self">Trace</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E1:92"><i><font color="Green">E49</font></i></a>, <a class="txt" href="cohsp_1.html#E3:92_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:92_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E3:92_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="cohsp_1.html#E6:92_1"><i><font color="Green">E52</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2"><b>now </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="E1:92_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:92_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E4:92_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="cohsp_1.html#V9" target="_self">U-stable</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:92_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K8" target="_self">Trace</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E1:92"><i><font color="Green">E49</font></i></a>, <a class="txt" href="cohsp_1.html#E3:92_2"><i><font color="Green">E52</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:92_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; ( 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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E3:92_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="cohsp_1.html#E6:92_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="cohsp_1.html#D18" target="_self">Def18</a></i>;<br/><a NAME="E9:92_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E10:92_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E3:92_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="cohsp_1.html#E6:92_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="cohsp_1.html#E8:92_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="cohsp_1.html#T34" target="_self">Th34</a></i>;<br/><b>hence </b><a NAME="E11:92_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K1">`1</a>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E8:92_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:92_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:92_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:92_3"/><i><font color="Green">E54</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>assume </b><a NAME="E5:92_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:92_3"/><i><font color="Green">E55</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being  <a href="cohsp_1.html#V9" target="_self">U-stable</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:92_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K8" target="_self">Trace</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E1:92"><i><font color="Green">E49</font></i></a>, <a class="txt" href="cohsp_1.html#E4:92_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="cohsp_1.html#E7:92_3"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E10:92_3"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E11:92_3"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E1:92_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="cohsp_1.html#E4:92_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="cohsp_1.html#E7:92_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="cohsp_1.html#E9:92_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="cohsp_1.html#T35" target="_self">Th35</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:92_4"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:92_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:92_4"/><i><font color="Green">E54</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>assume </b><a NAME="E5:92_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:92_4"/><i><font color="Green">E55</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being  <a href="cohsp_1.html#V9" target="_self">U-stable</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:92_4"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K8" target="_self">Trace</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E1:92"><i><font color="Green">E49</font></i></a>, <a class="txt" href="cohsp_1.html#E4:92_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="cohsp_1.html#E7:92_4"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E10:92_4"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E11:92_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E1:92_4"><i><font color="Green">E53</font></i></a>, <a class="txt" href="cohsp_1.html#E4:92_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="cohsp_1.html#E7:92_4"><i><font color="Green">E55</font></i></a>, <a class="txt" href="cohsp_1.html#E9:92_4"><i><font color="Green">E56</font></i></a>, <a class="ref" href="cohsp_1.html#T36" target="_self">Th36</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:92"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="cohsp_1.html#V9" target="_self">U-stable</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="cohsp_1.html#K8" target="_self">Trace</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E2:92"><i><font color="Green">E50</font></i></a>, <a class="txt" href="cohsp_1.html#E3:92"><i><font color="Green">E51</font></i></a>, <a class="txt" href="cohsp_1.html#E4:92"><i><font color="Green">E52</font></i></a>, <a class="ref" href="cohsp_1.html#T39" target="_self">Th39</a></i>;<br/>


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