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

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>();<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#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) );<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:69_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:69_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="catalg_1.html#T24" target="_self">Th24</a></i>;<br/><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>);<br/><b>thus </b><a NAME="E4:69_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>]
 <b>by </b><i><a class="txt" href="catalg_1.html#E3:69_1_1"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:69_1"/><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 <b>and </b><br/><a NAME="E4:69_1"/><i><font color="Green">E42</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> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</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>2</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="catalg_1.html#E1:69_1"><i><font color="Green">E40</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">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>) st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) ) ) or ( <font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2 &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>4</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)</span><a href="finseq_1.html#K10">*&gt;</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>) st <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) &amp; <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>) holds <br/><font color="Olive">b<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>5</sub></font>) ) ) ) ) );<br/>
<a NAME="E5:69_1"/><i><font color="Green">E43</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> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</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>2</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="69_1_2"><b>proof </b></a><div class="add">

<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="E1:69_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 ;<br/>

<a NAME="E2:69_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="catalg_1.html#T25" target="_self">Th25</a></i>;<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:69_1_2_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="catalg_1.html#E1:69_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="catalg_1.html#T26" target="_self">Th26</a></i>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <a href="xboole_0.html#K1">{}</a>  <a href="cat_1.html#K7">.--&gt;</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>4</sub></font>);<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <a href="xboole_0.html#K1">{}</a>  <a href="cat_1.html#K7">.--&gt;</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>4</sub></font>) as    <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E5:69_1_2_1"/><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="cat_1.html#K7">.--&gt;</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>4</sub></font>)</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E6:69_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="catalg_1.html#E3:69_1_2_1"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E7:69_1_2_1"/>
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>8</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font>)
 <b>by </b><i><a class="txt" href="catalg_1.html#E2:69"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E8:69_1_2_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>8</sub></font>)</span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font>)
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E9:69_1_2_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="cat_1.html#K7">.--&gt;</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>4</sub></font>)</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font>)
 <b>by </b><i><a class="txt" href="catalg_1.html#E8:69_1_2_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <a href="xboole_0.html#K1">{}</a>  <a href="cat_1.html#K7">.--&gt;</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>4</sub></font>) as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font>) <b>by </b><i><a class="txt" href="catalg_1.html#E5:69_1_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font>;<br/><a NAME="E11:69_1_2_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E12:69_1_2_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>8</sub></font>) )
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_1_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:69_1_2_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="catalg_1.html#E1:69_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="catalg_1.html#T27" target="_self">Th27</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</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="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>) &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>) &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>) );<br/><div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_2_1"><b>now </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="E1:69_1_2_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>)</span><a href="finseq_1.html#K10">*&gt;</a></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="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:69_1_2_2_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>) &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>) &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="funct_6.html#T2">FUNCT_6:2</a></i>;<br/><b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>);<br/><a NAME="E4:69_1_2_2_1"/>
( <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>by </b><i><a class="txt" href="catalg_1.html#E1:69"><i><font color="Green">E37</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_2_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>)
 <b>by </b><i><a class="txt" href="catalg_1.html#E3:69"><i><font color="Green">E39</font></i></a>, <a class="txt" href="catalg_1.html#E3:69_1_2_2_1"><i><font color="Green">E49</font></i></a></i>;<br/><b>thus </b><a NAME="E6:69_1_2_2_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font>]
 <b>by </b><i><a class="txt" href="catalg_1.html#E3:69_1_2_2_1"><i><font color="Green">E49</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="E6:69_1_2_2"/><i><font color="Green">E49</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>)</span><a href="finseq_1.html#K10">*&gt;</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>) )
 <b>and </b><br/><a NAME="E7:69_1_2_2"/><i><font color="Green">E50</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>  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>)</span><a href="finseq_1.html#K10">*&gt;</a></span> holds <br/><font color="Maroon">S<sub>3</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="wellord2.html#S1">WELLORD2:sch 1</a>(<a class="txt" href="catalg_1.html#E4:69_1_2_2"><i><font color="Green">E48</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>thus </b><a NAME="E9:69_1_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="catalg_1.html#E3:69_1_2_2"><i><font color="Green">E47</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>)</span><a href="finseq_1.html#K10">*&gt;</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font>) <b>by </b><i><a class="txt" href="catalg_1.html#E6:69_1_2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><b>take </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font>;<br/><b>thus </b><a NAME="E11:69_1_2_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E12:69_1_2_2"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>) &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>) )
 ;<br/><a NAME="E13:69_1_2_2"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>)</span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="funct_6.html#T2">FUNCT_6:2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:69_1_2_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>) &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>) &amp; <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>17</sub></font>) )
 <b>by </b><i><a class="txt" href="catalg_1.html#E7:69_1_2_2"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E16:69_1_2_2"/>
( <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E15:69_1_2_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E17:69_1_2_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font>)
 <b>by </b><i><a class="txt" href="catalg_1.html#E15:69_1_2_2"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:69_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( ( <font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Olive">b<sub>2</sub></font> &amp; ex <font color="Olive">b<sub>3</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font>) st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">b<sub>2</sub></font>) ) ) ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2 &amp; ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> &amp; ex <font color="Olive">b<sub>5</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>)</span><a href="finseq_1.html#K10">*&gt;</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>) st <br/>( <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>) &amp; <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>) holds <br/><font color="Olive">b<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>6</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>6</sub></font>) ) ) ) ) )
 <b>by </b><i><a class="txt" href="catalg_1.html#E2:69_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="catalg_1.html#E3:69_1_2"><i><font color="Green">E46</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:69_1"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 <b>and </b><br/><a NAME="E8:69_1"/><i><font color="Green">E45</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> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</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="catalg_1.html#E5:69_1"><i><font color="Green">E43</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <b>by </b><i><a class="txt" href="catalg_1.html#E3:69_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <b>by </b><i><a class="txt" href="catalg_1.html#E7:69_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E11:69_1"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E1:69_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> ;<br/>
<i><font color="Green">E46</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2">Arity</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>

<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D19">PBOOLE:def 19</a></i>
;<br/>
<i><font color="Green">E47</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>

;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_3_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_3_3"/>
( ( <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2 &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 ) )
 </span><b>by </b><i><a class="ref" href="catalg_1.html#T25" target="_self">Th25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_3_3_1"><b>suppose </b></a><a NAME="E1:69_1_3_3_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:69_1_3_3_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">c<sub>8</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font>) st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>8</sub></font>) ) )
 <b>by </b><i><a class="txt" href="catalg_1.html#E8:69_1"><i><font color="Green">E45</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E5:69_1_3_3_1"/><i><font color="Green">E49</font></i>: 
(  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>) )
 <b>by </b><i><a class="txt" href="catalg_1.html#E4:69_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E6:69_1_3_3_1"/>
(  <a href="msualg_1.html#K1">the_arity_of</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E5:69_1_3_3_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="catalg_1.html#T22" target="_self">Th22</a>, <a class="ref" href="catalg_1.html#T28" target="_self">Th28</a>, <a class="ref" href="relat_1.html#T62">RELAT_1:62</a></i>;<br/><b>hence </b><a NAME="E7:69_1_3_3_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="relset_1.html#M2">Relation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="catalg_1.html#E3:69_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="catalg_1.html#E4:69_1_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="catalg_1.html#E3:69_1_3_3_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="catalg_1.html#E5:69_1_3_3_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="card_3.html#T19">CARD_3:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_3_3_2"><b>suppose </b></a><a NAME="E1:69_1_3_3_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2 &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:69_1_3_3_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>)</span><a href="finseq_1.html#K10">*&gt;</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>) st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>) &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font>) ) ) )
 <b>by </b><i><a class="txt" href="catalg_1.html#E8:69_1"><i><font color="Green">E45</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E5:69_1_3_3_2"/><i><font color="Green">E49</font></i>: 
(  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>) )
 <b>by </b><i><a class="txt" href="catalg_1.html#E4:69_1"><i><font color="Green">E42</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E7:69_1_3_3_2"/><i><font color="Green">E50</font></i>: 
(  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>) )
 <b>by </b><i><a class="txt" href="catalg_1.html#E4:69_1"><i><font color="Green">E42</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E9:69_1_3_3_2"/><i><font color="Green">E51</font></i>: 
(  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>) )
 <b>by </b><i><a class="txt" href="catalg_1.html#E4:69_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E10:69_1_3_3_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="msualg_1.html#K1">the_arity_of</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E5:69_1_3_3_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="catalg_1.html#E7:69_1_3_3_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="catalg_1.html#E9:69_1_3_3_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="catalg_1.html#T22" target="_self">Th22</a>, <a class="ref" href="catalg_1.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E11:69_1_3_3_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>)</span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>) )
 <b>by </b><i><a class="txt" href="catalg_1.html#E3:69_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="catalg_1.html#E3:69_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="catalg_1.html#E4:69_1_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="catalg_1.html#E3:69_1_3_3_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="catalg_1.html#E5:69_1_3_3_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="catalg_1.html#E7:69_1_3_3_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="catalg_1.html#E9:69_1_3_3_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_2.html#T145">FINSEQ_2:145</a></i>;<br/><b>hence </b><a NAME="E12:69_1_3_3_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="relset_1.html#M2">Relation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="catalg_1.html#E3:69_1_3_3_2"><i><font color="Green">E48</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> ;<br/>
<b>take </b><font color="Maroon">c<sub>7</sub></font> =  <a href="msualg_1.html#G3">MSAlgebra</a>(# <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> #);<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E2:69_1_4"/><i><font color="Green">E46</font></i>: 
(  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>) )
 <b>by </b><i><a class="txt" href="catalg_1.html#E4:69_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E3:69_1_4"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E2:69_1_4"><i><font color="Green">E46</font></i></a>, <a class="ref" href="catalg_1.html#T22" target="_self">Th22</a></i>;<br/><b>hence </b><a NAME="E4:69_1_4"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>)
 <b>by </b><i><a class="txt" href="catalg_1.html#E2:69_1_4"><i><font color="Green">E46</font></i></a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><a NAME="E1:69_1_5"/>
( <span class="p1">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:69_1_5"/><i><font color="Green">E46</font></i>: 
(  <a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">c<sub>9</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font>) st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>9</sub></font>) ) )
 <b>by </b><i><a class="txt" href="catalg_1.html#E8:69_1"><i><font color="Green">E45</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  = 
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>9</sub></font>)
<b>by </b><i><a class="txt" href="catalg_1.html#E3:69_1_5"><i><font color="Green">E46</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>8</sub></font>)
<b>by </b><i><a class="txt" href="catalg_1.html#E3:69_1_5"><i><font color="Green">E46</font></i></a>, <a class="ref" href="catalg_1.html#T21" target="_self">Th21</a></i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>;<br/>
<a NAME="E15:69_1"/>
( <span class="p1">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2 &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E17:69_1"/><i><font color="Green">E46</font></i>: 
(  <a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>)</span><a href="finseq_1.html#K10">*&gt;</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font>) st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>) &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font>) ) ) )
 <b>by </b><i><a class="txt" href="catalg_1.html#E8:69_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E18:69_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="txt" href="catalg_1.html#E17:69_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="catalg_1.html#T23" target="_self">Th23</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>)</span><a href="finseq_1.html#K10">*&gt;</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>)<b> such that </b><br/><a NAME="E20:69_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; ( 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">F<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>) &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>) holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>) ) )
 <b>by </b><i><a class="txt" href="catalg_1.html#E17:69_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>

<b>assume </b><a NAME="E21:69_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>) &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>) )
 ;<br/>

<a NAME="E22:69_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>16</sub></font>)
 
<b>by </b><i><a class="txt" href="catalg_1.html#E20:69_1"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E23:69_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>16</sub></font>)
 <b>by </b><i><a class="txt" href="catalg_1.html#E20:69_1"><i><font color="Green">E47</font></i></a></i>;<br/>


</div>
