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

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>, B is    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="commacat.html#K3" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="commacat.html#K4" target="_self">`12</a>  </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="commacat.html#K5" target="_self">*</a> <font color="Olive">b<sub>2</sub></font> );<br/>
<a NAME="E1:8_1_1"/><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> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>, B is    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="commacat.html#K3" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="commacat.html#K4" target="_self">`12</a>  </span>}</span>  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="8_1_1_1"><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> ;<br/>

<b>assume </b><a NAME="E1:8_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>, B is    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="commacat.html#K3" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="commacat.html#K4" target="_self">`12</a>  </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:8_1_1_1"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="commacat.html#K3" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="commacat.html#K4" target="_self">`12</a>  )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="commacat.html#K5" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E5:8_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="commacat.html#K5" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="commacat.html#E3:8_1_1_1"><i><font color="Green">E8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:8_1_1"/><i><font color="Green">E8</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>, B is    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="commacat.html#K3" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="commacat.html#K4" target="_self">`12</a>  </span>}</span>  &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> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>, B is    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="commacat.html#K3" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="commacat.html#K4" target="_self">`12</a>  </span>}</span>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</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="zfrefle1.html#S1">ZFREFLE1:sch 1</a>(<a class="txt" href="commacat.html#E1:8_1_1"><i><font color="Green">E7</font></i></a>);<br/></i>
<a NAME="E4:8_1_1"/><i><font color="Green">E9</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:8_1_1_2"/>
<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>
 ;<br/>

<a NAME="E2:8_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> st <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="commacat.html#K3" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="commacat.html#K4" target="_self">`12</a>  )
 
<b>by </b><i><a class="txt" href="commacat.html#E3:8_1_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:8_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:8_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:8_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</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="E3:8_1_1_3"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>9</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>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:8_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> st <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="commacat.html#K5" target="_self">*</a> <font color="Olive">b<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="commacat.html#E3:8_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="commacat.html#E3:8_1_1_3"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:8_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="commacat.html#E3:8_1_1_3"><i><font color="Green">E10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>, <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="commacat.html#E4:8_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E7:8_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>, B is    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="commacat.html#K3" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="commacat.html#K4" target="_self">`12</a>  </span>}</span> 
 <b>by </b><i><a class="txt" href="commacat.html#E3:8_1_1"><i><font color="Green">E8</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>



<b>assume </b><a NAME="E8:8_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E10:8_1_1"/><i><font color="Green">E10</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="commacat.html#K5" target="_self">*</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="commacat.html#E3:8_1_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E11:8_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="commacat.html#E10:8_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E12:8_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="commacat.html#K5" target="_self">*</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="commacat.html#E10:8_1_1"><i><font color="Green">E10</font></i></a></i>;<br/>


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